The example you send has this menu added to the sidebar and it will not work on pages without sidebar like this
https://gyazo.com/861ff24001cb3afaf5f8c503666218bb

or with page heading block like this https://gyazo.com/956dd86332de6bf8a5cbba006aade4e0

Here is a solution we can propose you:

1. Disable option to always show this categories dropdown on the shop and single product page.

2. Add Vertical mega menu widget to your shop and product pages sidebar in Appearance -> Widgets and then it will be always opened.

And you will be able to keep your header menu opened on the home page only.

Kind Regards