Sort Posts with custom meta key by default which is currently set as optional

the change you require is a modification to your theme, it will take time and you might end up corrupting it if you are not comfortable working with WordPress, instead of that you can make a child theme for it and make whatever change you want inside it.

A replacement and partial solution would be to add a js in your theme, to click on the link as soon as the div is there, it will refresh the posts automatically for you.

jQuery(document).ready(function(){
 if( jQuery('.home.blog').find('.articles .icon-liked.liked ') != ''){
   jQuery('.home.blog').find('.articles .icon-liked.liked ').click();
   jQuery('.home.blog').find('.articles .sortbar-title').html('MOST LIKED');
 }
});

Also I did modified your theme.php to include script in page, added 

<script type="text/javascript" src="https://wordpress.stackexchange.com/questions/121647/<?php echo THEME_JS_URI; ?>/custom.js"></script>

to function 

 it_footer_scripts()

its merely a solution but its a solution as of now.

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