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I had the same experience with my MacPro with an ATI Radeon 5770. I replaced it by the 5870, where the problem didn't occur.
But perhaps you can try to flash your machine's firmware from 4,1 to 5,1.
The ATI Radeon 5770 was fully compatible with the 5,1 computers (2010).

Unfortunately that probably will not help as I have a 2010 running 5,1 firmware and a GTX 680 and it happens to me as well.
One thing I notice is if you constantly just restart (ie. Into Windows then back to OS X) it does not occur. A cold shutdown and boot and the problem will surface.
It seems a bunch of replacement cards are doing this. It isn't just 1 brand or model:
https://forums.macrumors.com/threads/1566964/
 
I think there is something about the sensors. If you do a cold boot, usually the fans will rev up. If you stress it out then the sensors will say "OH! there is a card there and take control. That is just putting it simply.
 
Unfortunately that probably will not help as I have a 2010 running 5,1 firmware and a GTX 680 and it happens to me as well.
One thing I notice is if you constantly just restart (ie. Into Windows then back to OS X) it does not occur. A cold shutdown and boot and the problem will surface.
It seems a bunch of replacement cards are doing this. It isn't just 1 brand or model:
https://forums.macrumors.com/threads/1566964/

what is your SMC version?
 
An Apple genius will just say the card is unsupported, or if you're unlucky, he will try to make you buy a new logic board. This is an SMC bug that Apple refuses to fix. The only workaround is to stress the GPU sufficiently to bring the PCIe fan up to 2086 rpms or so, after which it will drop back to baseline and behave normally. This normal behavior will persist through sleep and restarts, but not shutdowns.

Some people don't notice the problem because they stress their Mac Pros immediately. Populating all the HDD bays and/or PCIe slots also seems to help jack the PCIe fan revs, while booting off an SSD exacerbates the problem.

Also, note that it is both the PCIe and PSU fans that are affected. Both rev high and then return to baseline once the ~2086 rpm threshold is reached.

Too bad Apple is so strapped for cash that they can't hire a part-timer to get their only tower to work right with the only video cards they sell for it - oh wait, they're the most profitable company in the world! This SMC bug just demonstrates how much they value their professional user base.
 
Whatever the reason is for the problem, running the Open GL test does the trick. I tried it on my 2009 Mac Pro having this same problem with the ATI 5770 and voilà, fan is running quietly at a normal speed. Just thought I'd throw that out there. Thanks for the tip!:)
 
I still can't get my PCIe + PSU fan to calm down (upgraded to a Radeon 580RX Pulse) no matter what I do:

- SMC reset > nope

- Go in to iStats menu and override the fan speeds manually (lower them to around 800) > nope, PCIe still around 1200rpm

- Go in to iStats menu and override the fan speeds manually (rev them up and let the system take over again after that) > nope, PCIe still around 1200rpm and even PSU stays around 1000rpm
 
I still can't get my PCIe + PSU fan to calm down (upgraded to a Radeon 580RX Pulse) no matter what I do:

- SMC reset > nope

- Go in to iStats menu and override the fan speeds manually (lower them to around 800) > nope, PCIe still around 1200rpm

- Go in to iStats menu and override the fan speeds manually (rev them up and let the system take over again after that) > nope, PCIe still around 1200rpm and even PSU stays around 1000rpm


I have had that same problem for months. I had 140.0.0.0 BootROM installed. After upgrading, four days ago, to 144.0.0.0 and Mojave 10.14.5 I don't need Mac fan Control anymore.
 
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