Eventual unimodality under free convolution powers

Let $\mu be a compactly supported probability measure on$\mathbb R. There is a number $T\geq 1$ depending on $\mu.

Eventual unimodality conjecture. The free convolution powers $\mu^{\boxplus s} are unimodal for every$s\geq T$.

The partial free convolution semigroup is available for all probability measures when $s\geq1$, so the question extends the preceding discussion from freely infinitely divisible measures to compactly supported measures. The supplied text gives no resolution or supporting theorem for this assertion.