This was a newer GE Monogram side-by-side unit that used an inverter compressor. I removed the inverter and used an ohmmeter to measure the resistance of each of the three windings. They were all very close to one another: about 6.8 ohms per winding. I concluded that the compressor was probably not at fault.
The inverter kind of burned or chemical-ish, and, when opened, showed an electrolytic capacitor had burst. I replaced the inverter with an overnight part from Amazon. It was the same price as your part, but with faster delivery.
I restarted the refrigerator, but the compressor became very hot and did not cool the refrigerator or freezer.
I vacuumed the condenser coils, which were buried in dust.
The signal from the mainboard to the inverter (red and white wires) measured less than 5 volts, so I replaced the mainboard with your part.
I restarted the refrigerator, which began working properly.
Total cost: about $360 (about $150 for each part, plus $20-$30 for shipping). Total downtime: several days, because I did a lot of other things that were not necessary, such as totally disassembling the refrigerator and freezer, checking each thermistor with ice water, etc.
I think the thermistors could have been checked much more easily, without anywhere as much work disassembling things, this way:
If the refrigerator and freezer have been set at room temperature for a while, you really don't need to disassemble the unit to access each of the four thermistors. Instead, you can unplug J1 from the mainboard and measure resistance from the common (pin 5) to each of the thermistors (pins 1 through 4). Just like the compressor coil resistance, the resistance of each of the thermistors should be quite close to one another, if they are all at the same temperature. Any significant outlier would need to be understood (is the freezer cooler than the refrigerator? etc.) and possibly replaced. All four of mine were each about 6200 ohms at about 68 degrees (my home's temperature). I wish I hadn't wasted so much time tearing everything apart and verifying each one measured about 16,300 ohms in ice water.
I also learned that you can enter diagnostic mode by simultaneously pressing refrigerator and freezer up and down temp buttons (all 4 buttons) at the same time. This was helpful, but unfortunately, the mainboard did not diagnose its own fault.