HDMI is a "legacy" interface designed for HD TV 1920 by 1080p at best. It has seen a number of attempts to extend it, but it is a real balancing act getting all the right parts to work together.
The output of the Mac Pro 2013 HDMI port is not suitable for extra-large sizes like what you are attempting. I recommend you use a ThunderBolt/Mini-DisplayPort to Dsiplayport input:
Input Connectors
2 x HDMI 2.0 (HDCP 2.2) (10-bit color @ 60Hz)
1 x DP 1.4 (HDCP 2.2) (10-bit color @ 60Hz)
5 x USB 3.0 Downstream port
2 x USB 3.0 Upstream port
1 x USB Type-C (Alternate mode with DP1.4, Power Delivery, and USB2.0) (8-bit color @ 60Hz)
The state-of-the-art interface is DisplayPort (family). It drops the HUGE amount of 'heartbeat' refresh data required by CRT displays and simply sends the changes. That alone helps it to run faster and your computer to run cooler. Screen refresh comes from the screen's own internal buffer, and can be above threshold of perception for flickering..
Once you get on a DisplayPort interface, there may be a bit of software fiddling you still need to do to get 5K on that particular display. It appears the screen descriptor technology has 'run out of gas' and Apple seems to be working (at what seems to Users like slow motion) to come up with permanent fix.