A finite-entropy separation of microstates and nonmicrostates free entropy

We answer the finite-entropy equality question for microstates and nonmicrostates free entropy negatively. We construct a bounded self-adjoint tuple X in a von Neumann algebra with faithful normal tracial state such that $-\infty <\chi (X)\le {\chi }^{\ast }(X)-\frac{1}{2}<\infty$. Here χ is the original microstates entropy with an operator-norm cutoff and a limsup over matrix sizes. The counterexample uses a large but fixed number of variables.