Hello,
You can add the Categories element in the Header Builder of your Site first: https://snipboard.io/d8oWkq.jpg
Then edit that Categories element, select your choice of Menu in it: https://snipboard.io/gUkXom.jpg
Then you need to edit your Homepage with simple Gutenberg edit > there under the Page settings options, you need to click on Header > in that you can find the Open categories menu toggle option > you need to enable it then due to this the Categories menu will remain open on the Homepage: https://snipboard.io/14JyjT.jpg
Hope this helps !
Best Regards.