Privacy blue dot appearing on secondary displays even when in full screen when connecting by screen sharing

I updated to macOS 15.1 last week, and noticed that while using full screen apps while your computer is being observed using screen sharing from another Mac that a blue dot will stay visible in the top right hand corner of the screen on all displays. This computer is being used as a presentation computer so it's attached to a projector which is also showing that blue dot.

I tried doing the procedure noted in this post to remove the privacy indicator that appears when using a camera, but that did not do the trick.

Hide privacy indicators on external displays in macOS - Apple Support (CA)