Log-unimodality for free positive multiplicative Brownian motion

We prove that the marginal law $\sigma _{t}\mathrel {\scriptstyle {\boxtimes }}\nu $ of free positive multiplicative Brow\-nian motion is log-unimodal for all $t \gt 0$ if $\nu $ is a multiplicatively symmetric log-unimodal distribution, and that $\sigma _{t}\mathrel {\scriptstyle {\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.