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<?php
 add_filter( 'get_site_icon_url', 'custom_site_icon_url', 10, 3 );

 function custom_site_icon_url( $url, $size, $blog_id ) {

     if ( $size >= 512 ) {
         $size_data = 'full';
     } else {
         $size_data = array( $size, $size );
     }

     if(
        /* 条件 */
     ) {
         // 固定ページのアイキャッチ画像を取得する
         $page_object = get_page_by_path( /* 固定ページのスラッグ */ );
         if( !empty( $page_object ) ) {
             if ( has_post_thumbnail( $page_object->ID ) ) {
                 $post_thumbnail_id = get_post_thumbnail_id( $page_object->ID );
                 $url = wp_get_attachment_image_url( $post_thumbnail_id, $size_data );
             }
         }

     }

     return $url;
 }