Kaplansky's conjecture for Type II$_1$ AW$^*$-factors

An AW$^*$-factor is a Type II$_1$ AW$^*$-algebra with trivial centre, and a W$^*$-factor is a Type II$_1$ W$^*$-algebra with trivial centre.

Kaplansky's conjecture. Every Type II$_1$ AW$^*$-factor is a W$^*$-factor.

A positive solution would imply that every Type II$_1$ AW$^*$-factor admits a faithful tracial state. The conjecture is longstanding and remains open; it is equivalent to the assertion that every $2$-quasitrace on a unital C$^*$-algebra is a trace.