Can a flashed Mac Pro 2009 (5,1) support macOS Mojave
Yes, got Mojave on my 2009 5,1 flashed with the High Sierra update this morning using the 6/18 Mojave download.

The Apple store wouldn't let me download Mojave directly to my 2009, had to download on another machine (a 10.13 Fusion VM) then create a bootable USB, boot my 2009 from the USB and install from there.