Unimodality under free multiplicative convolution on the unit circle

Let $bc$ and $bd$ be symmetric unimodal probability distributions on the unit circle $a4$. The operation $boxtimes$ denotes free multiplicative convolution.

Unimodality conjecture. If $bc$ and $bd$ are symmetric unimodal distributions on $a4$, then $bc\boxtimesbd$ is also unimodal.

The analogous result for classical convolution $circledast$ was proved immediately beforehand. The paper reports that no counterexample to the free multiplicative version had been found; its resolution is not supplied here.