Log-unimodality for free positive multiplicative Brownian motion

We prove that the marginal law $\sigma_{t}\boxtimes\nu$ of free positive multiplicative Brownian motion is log-unimodal for all $t>0$ if $\nu$ is a multiplicatively symmetric log-unimodal distribution, and that $\sigma_{t}\boxtimes\nu$ is log-unimodal for sufficiently large $t$ if $\nu$ is supported on a suitably chosen finite interval. Counterexamples are given when $\nu$ is not assumed to be symmetric or having a bounded support.