Conjectures on free infinite divisibility of beta, beta prime, and gamma distributions

Let $\bm{\beta}_{p,q}$ and $\bm{\beta}'_{p,q}$ denote the beta and beta prime distributions, respectively, and let $\bm{\gamma}_p$ denote the gamma distribution, with parameters in the positive reals. A probability distribution is freely infinitely divisible (FID) if it is infinitely divisible with respect to free convolution. The beta, beta prime, and gamma conjectures. The following assertions are conjectured:

1. $\bm{\beta}_{p,q}$ and $\bm{\beta}'_{p,q}$ are not FID for $p\in(\frac{1}{2},\frac{3}{2})$ and $q>0$.
2. $\bm{\beta}_{p,q}$ is FID if $(p,q)\in(0,\frac{1}{2}]\times[\frac{3}{2},2]$ or if $(p,q)\in[\frac{3}{2},2]\times(0,\frac{1}{2}]$.
3. $\bm{\gamma}_p$ is not FID for $p\in(\frac{1}{2},\frac{3}{2})$.

These claims extend known non-FID results for special beta parameter families and propose further parameter ranges of free infinite divisibility and non-divisibility. The source does not establish the assertions in general.