Voiculescu's question: do microstates and nonmicrostates free entropy agree when finite?

Claims a negative answer: a bounded self-adjoint tuple with $-\infty<\chi(X)\le\chi^*(X)-\tfrac12<\infty$, built as a small semicircular perturbation of free semicirculars tensored with a commuting discrete variable. It does NOT settle equality for every smaller number of variables (the example uses a large fixed $n$), and uses the limsup version of $\chi$; equality under the known convexity conditions is unaffected.

Claims a negative answer: a bounded self-adjoint tuple with $-\infty<\chi(X)\le\chi^*(X)-\tfrac12<\infty$, built as a small semicircular perturbation of free semicirculars tensored with a commuting discrete variable. It does NOT settle equality for every smaller number of variables (the example uses a large fixed $n$), and uses the limsup version of $\chi$; equality under the known convexity conditions is unaffected.