Large-time unimodality for stable convolutions

Let $S_t$ be the law at time $t$ of a classical or free $\alpha$-stable process, where $\alpha\in(0,2)$, and let $\mu$ be a probability measure. Assume that

$$\int_{\mathbb R}|x|^{2+\alpha}\,d\mu(x)<\infty.$$

Stable-process unimodality conjecture. The convolution $\mu\ast S_t$ is unimodal for sufficiently large $t>0$.

This generalizes the preceding conjecture from the specific process with Lévy law $L_t$ to classical and free stable processes. The claim is presented as an expected extension and remains open in the supplied text.