My unit is the GK3V w/motherboard revision: "GK1_v23 09.04.2020" but the fan concerns seem to span multiple models.
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As most of you have mentioned, my fan was also only kicking on under heavy loads for the mostpart, but does tend to kick on for only a few seconds then goes off and cycles back on repeatedly. My first observation upon disassembling my unit was that the heat sink on these contains a single extra square of aluminum that is thermal pasted to the bottom of the main heat sink w/fins and the fan/sheet metal upper enclosure housing, but the extra square does NOT actually make contact with the die of the CPU. This will cause very spikey temperatures and possibly erratic fan behavior if the sensor data is changing rapidly in response to the CPU's power demands.
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My first order of business was to improve thermal conductivity by replacing the small aluminum square stuck to the bottom of the main heatsink assembly with a thermal paste stamp. I popped the aluminum square off and replaced it with 2 (old) pure copper pennies that I lapped down to flat discs and then soldered together. The 2-stack of pennies was then adhered with 0.15mm thermal tape. I noticed the pennies are at risk of contacting resistors on the surface around the CPU die so I took a thin (0.5mm I think) thermal pad and cut a square out of the center so I could lay it over the resistors and not obstruct contact with the die's surface.
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I repeated the steps twice more and made 2 other 2-stacks of pennies to be used as heatsinks for the memory chips as well. Those too needed a bit of thermal padding to protect from any unwanted electrical contact. So after using thermal grease and checking clearance/fitment all around I mounted it up and the fit is perfect!
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My fan also does the buzzing/stuttering and will jitter in place or rotate only a couple of degrees extremely slowly. I noticed this while running a benchmark to heat my machine up and expecting to see the fan come on but it wasn't. Turns out the fan control tries to turn the fan on right around 80C. This is the time that it buzzes/stutters/barely spins less than 1/10 of a rotation and so does not actually come on. At first I thought I had broken my fan but I probably just never noticed this behavior without physically looking at the fan while also watching the CPU temps. Continuing the stress testing further, I noticed the fan finally spun up when my machine was in the 90C-95C range and continued to do so intermittently. Thankfully I saw this before buying a replacement fan! Finally, going further, as my unit was getting into the 95C-105C range (while torturing it with Prime95) the fan FINALLY kicked on and stayed on full time keeping my unit just below its thermal throttling limit, but not actually thermal throttling (it was holding a steady 2.40GHz on all 4 cores the entire time). After turning the heat down by implementing a 9W power limit using ThrottleStop, it is now hovering right around 80C while crunching Prime95. There was about a 5 or 10 minute heat-soaked cooldown period that kept the fan on steady but then it went right back to cycling on and off as expected.
TLDR:
The stock heatsink assembly can easily and cheaply be improved using cheap thermal pads, a couple of old copper pennies, a fine hand file, thermal tape, double stick mounting tape, and foil HVAC tape.
The fan control leaves a bit to be desired... It is not pushing enough juice to start the fan at the low end of its active thermal cooling range causing the fan to cycle intermittently at medium loads (my fan goes starts buzzing around , and keeps the fan on constant under heavy-extreme loads (this is good!). I have not yet found a way to take manual control of it, but that is next on my list.
FAN BEHAVIOR:
00C - 50C -- fan off/no buzzing? (not sure because my box idles in the high 50C range)
55C - 79C -- fan off, but buzzing still if you listen closely or put your finger on it to feel the vibration, may jitter in place occasionally
80C - 94C -- this is the temperature range my fan actually starts kicking on intermittently
95C - 96C -- I noticed my fan still cycles intermittently but seems to stay on a little bit longer and spin a little faster
97C - 105C -- my fan stays on constantly at max speed after sustaining this temperature for a bit and will continue for a good 5-10 minutes after eliminating the load
Aside from BIOS / thermal sensors / EC settings, I imagine Intel's DPTF may also play a role in some of this behavior. Regardless though, you can boot this motherboard with no heatsinking at all and it will peak out at 105C very easily, but Intel's algorithms do keep the CPU safe and working by throttling it as needed.
In the end, I have a 10W TDP machine that can crank out 10W in a synthetic load all day long, albeit a bit toasty after 45 minutes+ but under normal usage conditions this thing does great.
EDIT:
This is the double stick mounting tape I used:
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pour mieux réguler la vitesse du ventilateur 
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