The AW*-factor problem for type II$_1$ factors

Let an AW-algebra* be a unital C*-algebra whose left annihilator of every nonempty subset is a principal right ideal generated by a projection. An AW*-factor is an AW*-algebra that is a factor, and a type $\mathrm{II}_1$ AW*-factor is a finite AW*-factor of type $\mathrm{II}_1$. A von Neumann factor is a factor that is also a von Neumann algebra.

AW-factor conjecture.* Every AW*-factor of type $\mathrm{II}_1$ is a von Neumann factor.

This is presented as a longstanding problem in the theory of AW*-algebras. The source does not state a resolution, so the problem remains open.