Making permalink for custom post type/custom taxonomy. Stuck with 404 in the single post

Question

I have looked around noth in StackOverflow and WP.SE.

Like many before me, I just want this structure

mysite.com/custom-post-type/custom-taxonomy-term/post-name

Or, in other words,

mysite.com/publication/book/post-name (single post)
mysite.com/publication/book/          (all posts in the term 'book')
mysite.com/publication/               (all posts in this publication CPT)

I followed this instruction by TheDeadMedic for making custom post type with custom taxonomy permalink, and another one by Matthew Boynes to fix the pagination issue.

All works well, except... everytime I open the single post, it returns the 404 page.

I have tried opening the Permalink page and setting flush_rewrite_rules() to no avail. I still can't open the single page.

The only fix that works is if I change the slug in my custom taxonomy to its original form.

So instead of publication (the rewrite slug for this tax, in accordance to my custom post type name), I would have the default type (the default slug for this tax). Which would defeat the purpose.

Here is my code.

Taxonomy.

    function cptui_register_my_taxes_type() {

/**
 * Taxonomy: Types.
 */

$labels = array(
    "name" => __( 'Types', 'mysite' ),
    "singular_name" => __( 'Type', 'mysite' ),
    "menu_name" => __( 'Types', 'mysite' ),
    "all_items" => __( 'All Types', 'mysite' ),
    "edit_item" => __( 'Edit Types', 'mysite' ),
    "view_item" => __( 'View Type', 'mysite' ),
    "update_item" => __( 'Update Type Name', 'mysite' ),
    "add_new_item" => __( 'Add New Type', 'mysite' ),
    "new_item_name" => __( 'New Type Name', 'mysite' ),
    "parent_item" => __( 'Parent Type', 'mysite' ),
    "parent_item_colon" => __( 'Parent Type:', 'mysite' ),
    "search_items" => __( 'Search Types', 'mysite' ),
    "popular_items" => __( 'Popular Types', 'mysite' ),
    "separate_items_with_commas" => __( 'Separate Types with commas', 'mysite' ),
    "add_or_remove_items" => __( 'Add or remove Types', 'mysite' ),
    "choose_from_most_used" => __( 'Choose from the most used Types', 'mysite' ),
    "not_found" => __( 'No Types found', 'mysite' ),
    "no_terms" => __( 'No types', 'mysite' ),
    "items_list_navigation" => __( 'Types list navigation', 'mysite' ),
    "items_list" => __( 'Types list', 'mysite' ),
);

$args = array(
    "label" => __( 'Types', 'mysite' ),
    "labels" => $labels,
    "public" => true,
    "hierarchical" => true,
    "label" => "Types",
    "show_ui" => true,
    "show_in_menu" => true,
    "show_in_nav_menus" => true,
    "query_var" => true,
    "rewrite" => array( 'slug' => 'publication', 'with_front' => true,  'hierarchical' => true, ),
    "show_admin_column" => true,
    "show_in_rest" => false,
    "rest_base" => "",
    "show_in_quick_edit" => true,
);
register_taxonomy( "tyoe", array( "publication" ), $args );
}

add_action( 'init', 'cptui_register_my_taxes_type' );

Custom Post Type

function cptui_register_my_cpts_publication() {

    /**
     * Post Type: Publication.
     */

    $labels = array(
        "name" => __( 'Publication', 'mysite' ),
        "singular_name" => __( 'Publication', 'mysite' ),
        "menu_name" => __( 'Publication', 'mysite' ),
        "all_items" => __( 'All Publication', 'mysite' ),
        "add_new" => __( 'Add New', 'mysite' ),
        "add_new_item" => __( 'Add New Publication', 'mysite' ),
        "edit_item" => __( 'Edit Publication', 'mysite' ),
        "new_item" => __( 'New Publication', 'mysite' ),
        "view_item" => __( 'View Publication', 'mysite' ),
        "view_items" => __( 'View Publications', 'mysite' ),
        "search_items" => __( 'Search Publications', 'mysite' ),
        "not_found" => __( 'No publication found', 'mysite' ),
        "not_found_in_trash" => __( 'No trashed publications', 'mysite' ),
        "parent_item_colon" => __( 'Parent Publication:', 'mysite' ),
        "featured_image" => __( 'Cover or illustration', 'mysite' ),
        "set_featured_image" => __( 'Set cover or illustration for this publication', 'mysite' ),
        "remove_featured_image" => __( 'Remove cover or illustration for this publication', 'mysite' ),
        "use_featured_image" => __( 'Use as cover or illustration for this publication', 'mysite' ),
        "archives" => __( 'Publication archives', 'mysite' ),
        "insert_into_item" => __( 'Insert into publication info', 'mysite' ),
        "uploaded_to_this_item" => __( 'Uploaded to publication info', 'mysite' ),
        "filter_items_list" => __( 'Filter publication', 'mysite' ),
        "items_list_navigation" => __( 'Publication list navigation', 'mysite' ),
        "items_list" => __( 'Publication list', 'mysite' ),
        "attributes" => __( 'Publication Attributes', 'mysite' ),
        "parent_item_colon" => __( 'Parent Publication:', 'mysite' ),
    );

    $args = array(
        "label" => __( 'Publication', 'mysite' ),
        "labels" => $labels,
        "description" => "",
        "public" => true,
        "publicly_queryable" => true,
        "show_ui" => true,
        "show_in_rest" => false,
        "rest_base" => "",
        "has_archive" => true,
        "show_in_menu" => true,
        "exclude_from_search" => false,
        "capability_type" => "post",
        "map_meta_cap" => true,
        "hierarchical" => false,
        "rewrite" => array( "slug" => "publication/%type%", "with_front" => false ),
        "query_var" => true,
        "menu_position" => 6,
        "menu_icon" => "dashicons-format-aside",
        "supports" => array( "title", "editor", "thumbnail", "comments", "revisions" ),
        "taxonomies" => array( "type" ),
    );

    register_post_type( "publication", $args );
}

add_action( 'init', 'cptui_register_my_cpts_publication' );

Setting up the permalink

function change_permalink_publication( $link, $post ) {
    if ( $post->post_type === 'publication' ) {
        if ( $terms = get_the_terms( $post->ID, 'type' ) )
            $link = str_replace( '%type%', current( $terms )->slug, $link );
    }

    return $link;
}

add_filter( 'post_type_link', 'change_permalink_publication', 10, 2 );


add_action( 'generate_rewrite_rules', 'change_permalink_publication_pagination' );
function change_permalink_publication_pagination( $wp_rewrite ) {
    unset($wp_rewrite->rules['publication/([^/]+)/page/?([0-9]{1,})/?$']);
    $wp_rewrite->rules = array(
        'publication/?$' => $wp_rewrite->index . '?post_type=publication',
        'publication/page/?([0-9]{1,})/?$' => $wp_rewrite->index . '?post_type=publication&paged=' . $wp_rewrite->preg_index( 1 ),
        'publication/([^/]+)/page/?([0-9]{1,})/?$' => $wp_rewrite->index . '?type=' . $wp_rewrite->preg_index( 1 ) . '&paged=' . $wp_rewrite->preg_index( 2 ),
    ) + $wp_rewrite->rules;
}

Is there a way to fix this?