Just for fun, I made a bunch of videos of KataGo's raw policy net output snapshotted over the course of the current run in a few different opening positions.
I didn't make a snapshot with every net, since that would quite expensive to load so many nets, so I just did a selection of them. The snapshots step exponentially through the run, so sampling every net early on, but then increasingly skipping wider and wider gaps later on. Each successive snap is 1.03x further in the run than the previous snap (except at the start, where it's always just the very next net). So for example, the entire final *half* of the run is compressed into merely the last 10ish seconds of the video.

I hope these links work!

Empty board: https://streamable.com/zvc4ms
Opening position 1: https://streamable.com/czs5fq
Opening position 2: https://streamable.com/ve37q4
Opening position 3: https://streamable.com/wa8duq
Opening position 4: https://streamable.com/dxyhny
Opening position 5: https://streamable.com/lpbsb0

I hope they're interesting!

on the "Empty board" you can see the progress towards the center. is it possible to move further with the advent of stronger networks? or is there any certainty that this will not happen?

No certainty, but I think it's unlikely.

Keep in mind the exponential scale - almost all the "progress" happens in like the first 1% of the run. After that there wasn't much change, except for tiny flickers at 4-5 sometimes.