My own report here regarding ASM1166 on PCB "PCIe 3.0 x4" so more likely same card as yours:
In reality it works as x2, not x4. My Linux Debian 12 system works as a C8 State. While connecting random PCIe cards it change away to usually C3, then I need to apply script to get C7, but I'm not able to achieve C8 with any PCIe card connected.
Now, I was positively surprised by this little ASM1166 because it started as C7 out of the box, I wasn't able get to C8, but C7 out of the box. I run this card on 221118-0000-00 firmware.
I was reading that sub versions: 221118-003E-00 or 221118-0048-00 may have some ASPM problems on some motherboards.