Mac Pro 4.1 (flashed) High Sierra Firmware Update
I found this on MacRumors...

Yeah you get a firmware update! :eek: first time the 5,1 has gotten a
firmware update in ages! I did not get it when updating from DP1 to
DP5 but if you make a DP5 installer, it says you must update your boot
rom before you can install and it tells you how, the shut down button
did not work for me but going to the apple menu and hitting shut down
worked for me :) then i followed the normal hold power button until
flashing power light and bam it did the firmware update! and im using
a flashed 4,1 to 5,1 Mac Pro! so yeah looks like flashed 4,1s are so
identicial they happily take the High sierra boot rom update! :D

So it looks like you'll be OK if you:

Click the shut down button which lets the installer write the firmware to disk.
Go to the installer menu and click &quot;Shut down&quot;.
After the machine shuts down, hold down the power button for 15+ seconds until you hear a beep. The power light will flash quickly a few times before it beeps
Release the power button and allow the firmware update to proceed. DO NOT shut down the machine during the firmware update.

Reading further down that thread -

Just a heads up to update the firmware on a Mac Pro 4,1 or 5,1 you
have to install an EFI video card (this is true also if your flashing
a 4,1 to a 5,1) if you dont have a EFI video card then it won't update

Note: I gave up after about 6 pages of that thread, but the general consensus seems to be the above instructions do work.
Let us know how it goes - I have a similar machine...
Update April 2018
I just attempted this myself [eventually]. I first stripped all except the boot drive [Samsung SSD in the optical bay] &amp; keyboard/mouse. Successfully got the Restart button to work, no probs, but the firmware update didn't succeed after holding the button until the light flashed.
After a bit of Googling, I found a note that said to hold the button not only until the light flashes, but keep holding until you hear the beep tone. After doing that, it worked &amp; all went smoothly after that. The OS update is actually a separate process, which I hadn't realised until that point.