

Would they do that? No idea. But could they? Yes, without a doubt. I don’t know that browser in particular, but if you’ve installed some software on your machine, that software can do anything any other software could do.
edit: this page (which seems like it might be from the developers of that browser) indicates it can monitor your screen and restrict your Internet access.
Or in a VM if you don’t have any spare devices available. VM escapes exist, but they’re a pretty rare and severe type of vulnerability that’s unlikely to be casually utilised by proctoring software.
With 2FA I probably wouldn’t have too much of a problem with doing this. Especially if I then change password afterwards.
Another option would be to host it somewhere that you can remember the URL. If you don’t care for the privacy of the document itself, just using a URL shortener and Google Drive’s public sharing would work fine, or hosting at your own domain.
Personally though, I’m glad that on the rare occasion I need to get something printed (I have my own black and white laser printer at home for 99% of my needs), my local company for that sort of thing lets you upload it from home and pick up.