Random Rodrigues descendants converge to deterministic measures

Let $\x03be$ be a probability measure compactly supported in $\x1d53d$, let $P_n=\x220f_{i=1}^n(x-\x03be_i)$ have roots independently sampled according to $\x03be$, and let $\x03bc_n$ be the root-counting measure of $Q_n=P_n^{(\x03b1_n)}$, where $\x03b1_n$ is a sequence of non-negative integers. Random-descendant convergence conjecture. (i) If $\frac{\alpha_n}{n}\to0$, then $\x007b\x03bc_n\x007d$ converges in probability to $\x03be$. (ii) If $\frac{\alpha_n}{n}\to\x03b1$ with $0<\x03b1<1$, then $\x007b\x03bc_n\x007d$ converges in probability to a measure $\x03be_\x03b1$ whose support is contained in the convex hull of the support of $\x03be$. This proposes a measure-theoretic extension of the paper's asymptotic formula from polynomial-root measures to arbitrary compactly supported probability measures; the source presents it as a guess, and no proof or resolution is supplied.