eventually I got it all in pieces. To cut a long story short here's the culprit :
No surprise here. Unfortunately the broken ring pieces have slightly scratched the cylinder surface
I have no idea how bad it is, really. To the touch it doesn't feel like much but I'd need a specialist to know more. I've called BMW to know what they would do but over here they're clueless (or they don't like people doing their own mechanics, whichever. Maybe both). For sure they don't carry oversize pistons so I guess the cylinders aren't supposed to be rebored ?
So has anybody direct experience with this, how does one fix the cylinder surface ?