I'm curious why your reported firmware is different than mine. – Mac Pro 5,1 with all High Sierra updates have been applied.
As John W alluded to, and as we've discussed in the classic Mac Pro topic, for Mac Pro 2009-2012, firmware (BootROM) updates are not automatically applied by system updates. You must run a full macOS installer, and it will then prompt you to manually update the BootROM before proceeding with the macOS install. This Macrumors thread has much more information about the various BootROM versions included with each macOS installer.There doesn't seem to be a way to force an update to the Boot ROM, and I thought an update was supposed to occur during the system upgrade, so I'm not sure what's going on.
But the short of it is that everyone should probably update to the last BootROM version available, 126.96.36.199, which is included with "Install macOS 10.14.5 (and 10.14.6) Mojave", regardless of what macOS version you intend to run. As John W said, you don't actually have to install Mojave. After the BootROM update, it will reboot and prompt you to begin the install Mojave process, at which point you can quit. You can continue to run any OS supported by your hardware, even after the BootROM update (that means as far back as Leopard, in most cases).
Note: if you haven't updated your BootROM before, and have something less than MP51.0089.B00, then you must run the Install macOS 10.13.6 High Sierra full installer first to update to BootROM MP51.0089.B00, and then you can run the Mojave full installer to get to 188.8.131.52.