I set aside some time today to do the firmware upgrade in the Mojave 10.14.6 full install. Being the cautious type, I removed all internal drives (except, of course, the High Sierra boot drive) before running the installer. No problems; it went quickly.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.
But the short of it is that everyone should probably update to the last BootROM version available, 18.104.22.168, 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).
BootROM on this Mac Pro 5,1 is now 22.214.171.124.0.
Thanks for the detailed explanation of why and when the firmware gets updated.
And I can still boot into Snow Leopard, Mountain Lion, and Sierra.