That part I don't know. I haven't seen a comparison between them, but it's compared well at least against BPG, or whatever the h264 based format was. I don't think i've seen a comparison against h265, let along AV1. It'd be interesting to see. Even if they're about as good, I like FLIF's setup better since it enables progressive decoding which could be a big step up for slower connections.
https://wyohknott.github.io/image-formats-comparison/report....
FLIF is reported at 59.57% avg space savings
AV1 (2018-02-22) is reported at 66.02% avg space savings