  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.
