Non-free-infinite-divisibility conjectures for powers of semicircular variables

Let $S$ have the centered semicircular distribution with variance parameter $1$, written $S\sim\mathcal{S}(0,1)$. Let $r$ be real. Semicircular power conjectures. The following assertions are expected:

1. If $r\in(0,2)$, then

$$|S|^r\not\sim {\rm FID}.$$

2. If $r\in(1,2)$, then

$$|S|^r\,\operatorname{sign}(S)\not\sim {\rm FID}.$$

These problems are posed to complete the analogy between the classical and free cases. The source gives them as problems remaining to be solved and provides no resolution.