The capacitors next no the other cache chip are identical to those you need to install on the empty pads.

But I can not give any gurantees. Everything I wrote is simply deducted from the pictures I found and I haven't got any schematics verify.

The board turning on on its own sounds dodgy still. 🙁
Is there a BIOS option for "restore power on AC power loss" or similar?
Edit: Found this in the manual:

The Soft-Off mode refers to powering off the system through a momentary button switch (ATX switch) or through the software as opposed to disconnecting the AC power by way of a rocker switch or other means.

So the board appears to have a setting that makes it turn on as soon as power is applied to the PSU instead of using the ATX switch. That would explain the instant power on after the PSU was connected to the board. On second thought I may be interpreting this wrong and it simply explains ATX power management since that's new at the time.
(I'm not sure I understood right: Did you plug the PSU in on the motherboard while the PSU was connected to mains power? It is good practice to disconnect the PSU from mains power before plugging it into the motherboard. Plugging the ATX connector into the motherboard with the PSU on can also in some cases trigger a system start since ATX PSUs always supply +5VSB.)

Doornkaat wrote on 2021-06-03, 08:51:

The board turning on on its own sounds dodgy still. 🙁

Actually, it is usually the case the ATX boards start up on AC connection. It's the responsibility of the BIOS to check whether the machine should keep running or not, so the AC power loss policy can be implemented in software. In the case of this thread, the addition of the cache chip seems to break the front-side bus, so the BIOS code won't get executed, and the power suppy keeps turned on. Without the cache module, the board very likely turns on for a couple of milliseconds when AC power is connected. This is short enough you don't notice any motors starting.

So in the end I opened a dispute on paypal.
After a day I received a refund for the fake chips.