I had a somewhat different blank screen when connecting to secondary machines problem.   In my case, I do NOT get a blank screen if the computer's own monitor is displaying an image.  If the machine's monitor has gone dark, so is the MP window on my primary computer.

I have found that pressing Ctrl+Alt (no other key) "wakes up" the secondary machine to the same screen I'd see on the machine's connected monitor.  Sometimes it takes 2 or 3 tries, but then, all is normal.

Surprisingly, clicking the Ctrl/Alt/Del button on the MP status bar does not "wake up" the machine but works just fine once I use Ctrl+Alt to wake it up.