I believe that it was an open question back when I was a graduate student whether every word hyperbolic group admits a finite complete (=Church-Rosser=Noetherian+confluent) rewriting system for some finite generating set.

I am basically wondering if this question is still open, and if not what is the counterexample. (I find it hard to believe that every word hyperbolic group does have a finite complete rewriting system but would be happy also if somebody has proven this.)

The only necessary conditions I know for having a finite complete rewriting system are solvable word problem and $FP_{\infty}$.