Kabluchko's almost-sure convergence conjecture for critical points

Let $\mu$ be a probability measure on $\mathbb{C}$, let $X_1,X_2,\ldots$ be independent and identically distributed with law $\mu$, and define $P_n(z)=\prod_{j=1}^n(z-X_j)$. Let

$$\mu_n^{(1)}=\frac{1}{n-1}\sum_{z\in\mathbb{C}:P_n'(z)=0}\delta_z$$

be the empirical probability measure of the critical points of $P_n$. Kabluchko's conjecture. For any probability measure $\mu$ on $\mathbb{C}$, the sequence $\mu_n^{(1)}$ converges to $\mu$ almost surely as $n{\tilde o}\infty$.

This strengthens convergence in distribution to almost-sure convergence. The source identifies it as a conjecture stated by Kabluchko at a 2021 workshop and gives no resolution.