Beta-gamma characterization conjecture for sums of i.i.d. random variables

Let $n\geqslant2$ and $\textbf{Y}=(Y_1,Y_2,\ldots,Y_n)$ be a vector of i.i.d. random variables. Let $G_a$ denote a gamma random variable with parameter $a$, and let $B_{a,b}$ denote a beta random variable with parameters $a,b$. For $V=B_{a,(n-1)a}$ independent of $\textbf{Y}$, the beta-gamma characterization conjecture. $Y_1$ is equal in distribution to $cG_a$ for some constant $c>0$ if and only if

$$Y_1\stackrel{d}{=}V(Y_1+Y_2+\ldots+Y_n).$$

The forward implication follows from beta-gamma algebra, and the converse is known when $a=1/(n-1)$ and under the assumption that $Y_1$ has all positive integer moments finite. The uniqueness of the relevant scale-family solution to the integral equation is open in general.