Perhaps something was lost in translation here. I had two firmware updates to do, the first was to install High Sierra and the preceding FW Update (this may have been a Sierra FW update, if there's such a thing; the newest OS installed I believe was Yosemite). Are you saying the ATI 5770 cannot update to High Sierra?
This is clearly explained in the first post.
- Any model of the AppleOEM GPUs is required for you to upgrade any of the earlier MacPro5,1 firmwares until you have MP51.0089.B00.
- High Sierra supported METAL GPU is required when you have MP51.0089.B00. Also if you have any of the earlier Mojave firmwares (138.0.0.0.0 to 142.0.0.0.0), a High Sierra METAL supported GPU is required for you to update to the current 144.0.0.0.0.