I would also like to determine if the signal saying stop is coming from the motherboard.
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or if the fan itself stops regardless of the fact the motherboard is sending the correct values
To know what's going on you must monitor several readings. You need to know what volt levels go to the fan. You must confirm temperature of the device (or more than one device) that causes the fan to turn on or off. It's a matter of setting up a laboratory around your computer, as it goes through many sessions of operation. Watching readings as it heats up, and as it cools down.
It's important to determine whether the problem is in the fan or the motherboard. I'd expect to spend several hours running one of them apart from the other.
Discover what the fan does under what conditions of temperature, or supply voltage, etc.
Discover how hot the motherboard gets in order for it to send a 'turn on fan' signal. Discover how cool it gets in order to send an 'Off' signal.
Direct a plain fan toward it to see what range of temperature you get from the onboard sensor. It's hard to be sure whether the enclosure needs to be in place while the computer is on. The enclosure may be designed to enhance air movement so it cools better.
Ideally I'd compare results with an identical new fan, in case my fan is faulty.
To tell the truth, I suspect I'd eventually break something on my expensive motherboard. And I'd say 'Why couldn't I leave well enough alone?' (which I've asked myself more than once in the past).
If you think the fault may be in the system bios, then an update may be available. You can see how involved all this gets.
It may be easier (per post #2) to install a different fan, or auxiliary fan.
Assume the fan is needed all the time the computer is turned on, and adjust its speed accordingly.