Hi, I juste look at your site, understand the request.
You can't achieve this behavior with Webflow only, using the Webflow DropDown Menu widget. A drop down can't be set open for production, the open menu button is made as a designer tool solely, I'm sure.
I don't immediately see a solution to hand-craft the menu without using the drop-down widget, have it in a symbol, and behaving like you want. (I could be wrong but even by using conditional visibility and CMS collections, I can't find a way for a menu to open depending on the page we're on.)
So, of course, it it possible to get the behvior you want if 1/ you hand craft the menu with handmade dropdowns and 2/ if the menu is not a symbol and... you know the downsides.
- you find a clever mind that is going to find how Webflow opens the drop-down and successfully triggers is with code (by triggering the actual trigger or... simulating a click, very easy in JS)
- you find any capable JS dev and have them develop the full menu, with your brief.