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We prove that the marginal law σt ν of free positive multiplicative Brownian motion is log-unimodal for all t > 0 if ν is a multiplicatively symmetric log-unimodal distribution, and that σt ν is log-unimodal for sufficiently large t if ν is supported on a suitably chosen finite interval. Counterexamples are given when ν is not assumed to be symmetric or having a bounded support. Zhong proved in [Zho15] that σ t ν is absolutely continuous with a con-doi:10.4064/cm8413-6-2021 fatcat:uo4tbfea6fho7dp6wgyevigfvq