The 2-quastrace conjecture

A $2$-quasitrace on a unital C$^*$-algebra is a quasitrace that extends to a quasitrace on its matrix algebra $M_2(\mathcal{A})$; a trace is a normalized positive linear functional satisfying $\tau(x^*x)=\tau(xx^*)$ for all $x$.

The 2-quastrace conjecture. Every $2$-quasitrace on a unital C$^*$-algebra is a trace.

Blackadar and Handelman proved that this assertion is equivalent to Kaplansky's conjecture. Haagerup proved it for exact C$^*$-algebras, but the general assertion remains open.