I used this website to check my IPv6 connectivity and at first checking/unchecking IPv6 in adapter settings did have the expected effect, the website did/didn't see my IPv6 address. However, for whatever reason, after a while I couldn't get IPv6 connectivity at all, regardless of adapter settings. I tried to restart the adapter a couple of times, but cannot get it to work again. Currently, both the website and adapter properties show that there is no IPv6 internet access, despite it being enabled in adapter settings. Only IPv4 still works.
So, perhaps the problem was only to do with Windows AutoTuning after all. I will try to test it more in the next couple of days. Fortunately, my one file transfers are still good.