No where does it say you need an entry stamp in your passport. If this concerns ways to prove your current location, then there are plenty of other options to use.
That's incredible. Over the past few weeks we've seen posts of people being approved by the LA consulate in record time. It started with a few days, then a few hours, now less than an hour.
You can work remotely anytime even on visa exemption or a regular tourist visa. I mean, if you're in your hotel room or condo and doing some work for your employer on your computer: what's wrong with that? Who's gonna stop you? Heck you can even take your laptop to a cafe, sit there and do something on your computer. Again: what's wrong with that?