FYI, the Late-2013 Mac Pro (aka Trash Can) has known GPU failures. Apple had a free repair program for a small number of them back in the day, but my most of my organization's Late-2013 Mac Pros had confirmed GPU failures (GPU related Kernel Panics) which were not covered by the Apple free repair program which indicates Apple's estimate of affected products was completely inaccurate including GPUs which Apple said were never part of the program.
In addition, there are two GPU boards in this Mac which work in tandem (each board does something a bit different). Either one, or even both of them could be bad. There is no way to know. Even Apple authorized techs had no clue since Apple never provided any way of confirming except by spending money replacing one, then the other board.
After all this time, I guess you could have a bad internal connection. Reseating the boards & cables may freshen up the internal connections. However, this specific computer is not easy to work on since it does not sit flat on a work bench. You must basically disassemble the whole device just to access the GPU boards with it rolling around. The connections used are also very fragile & easily damaged.....absolute terrible design. And some of the screw/bolts can easily be stripped during reassembly.
From what I saw with my organization's Late-2013 Macs......you should save your money and retire it. It
Just curious if you tried a PRAM Reset (hold for at least three chimes)?
Also, make sure to disconnect all Thunderbolt devices since two of those Thunderbolt ports (I forget which ones) share the bus with the HDMI Port and can easily interfere with one another.
