Initial laws with all moments that stay multimodal under stable evolution

Hasebe and Ueda conjectured that if $\mu$ is a probability measure with finite moment of order $2+\alpha$, then the law at time $t$ of a classical, or free, strictly $\alpha$-stable process started from $\mu$ is unimodal for all sufficiently large $t$. We show that the conjecture fails in both settings for the positive stable processes of index $1/2$. In each case we give an initial law with finite moments of every order whose evolved law is not unimodal at any time $t>0$. In the classical case the initial law is supported on a sparse sequence of points going to $-\infty$, and near each of them the one-sided kernel cannot see the heavier mass lying to its right. In the free case the subordination function of the free convolution, analysed with the global inversion theorem of Huang and Wang, produces an interval on which the density vanishes, with positive density on both sides of it.