Tracial quotient of a free product versus free product of tracial quotients

Determine whether, for arbitrary unital separable C*-algebras A1 and A2 with nonempty trace spaces, the tracial quotient of their full free product equals the full free product of their tracial quotients; equivalently, establish whether Atr is canonically ∗-isomorphic to A1,tr ∗ A2,tr for A := A1 ∗ A2.

Background

For a unital separable C*-algebra A with nonempty trace space, its tracial quotient Atr is the largest quotient admitting a faithful tracial state, and the trace simplex T(A) is affinely homeomorphic to T(Atr). For a full free product A = A1 ∗ A2, every tracial representation of A factors through A1,tr ∗ A2,tr, giving a natural surjective ∗-homomorphism A1,tr ∗ A2,tr → Atr.

The question asks whether this canonical surjection is always an isomorphism. It is known to hold when A1,tr and A2,tr are residually finite dimensional (RFD), but the general case remains unresolved. Equivalently, this asks whether the free product of two unital separable C*-algebras with faithful tracial states necessarily admits a faithful tracial state.