The Dirichlet-weighted Arcsin conjecture for power semicircle distributions

Let $n\geq 2$, let ${\bf R}=(R_1,\ldots,R_n)$ have the Dirichlet distribution $D_n(1,\ldots,1)$, and let $X_1,\ldots,X_n$ be independent random variables, independent of ${\bf R}$, each having the Arcsin distribution on $(-a,a)$. Define

$$S_n=\langle {\bf R},{\bf X}\rangle=\sum_{i=1}^nR_iX_i.$$

Dirichlet-weighted Arcsin conjecture. The random variable $S_n$ has the power semicircle distribution on $(-a,a)$ with $\lambda=\frac{n-1}{2}$ and density

$$f(x;\lambda,a)=\frac{1}{\sqrt{\pi}a^{2\lambda}}\frac{\Gamma(\lambda+1)}{\Gamma\left(\lambda+\frac{1}{2}\right)}(a^2-x^2)^{\lambda-\frac{1}{2}},\qquad |x|<a.$$

The conjecture concerns a generalization of unimodality obtained by taking a Dirichlet-weighted inner product of independent random variables. The cited conclusions are presented as supporting context, but the supplied material does not establish whether the conjecture has been proved or remains open.