Superfish main menu children are breaking on certain pages

Preface

I have a (Superfish JS) horizontal main menu in my Drupal 7.38 site. Some of it's links being opened for sub menus, and These sub menus contains the children of the parent links.

This menu appears in the header section of the site, and in all pages.

The problem

When I surf from Mozilla firefox 39.0 and visit my home page, the triggered sub-menu appears to break, but when I surf that way in another page of the menu, it doesn't break...

Here is a live example.

My final aim

1. A plausible way to determine if the problem is JS or CSS based.
2. A detailed suggestion for a solution (especially if it's JS based since I'm just starting to learn JS and I almost know nothing about it).

Would affirm & rate if working. Thank you!

For some mysterious reason, after clearing Drupal cache, and rightafterwords - Cloudflare cache, the problem seem to no longer occur.