Hydrodynamic limit conjecture for roots of repeatedly differentiated random polynomials

Let $(\xi_j)_{j\geq 1}$ be independent complex-valued random variables with distribution $\mu$, let $P_n(z)=\prod_{j=1}^n(z-\xi_j)$, and let $\mu\lsem P_n^{(k)}\rsem$ denote the empirical probability measure of the roots of the $k$-th derivative. Fix $0\leq t<1$, let $k=k(n)$ satisfy

$$\frac{k}{n}\to t\qquad(n\to\infty),$$

and suppose that $\mu$ is compactly supported. Hydrodynamic limit conjecture. The random probability measure $\mu\lsem P_n^{(k)}\rsem$ converges, in the stochastic or even almost-sure sense as an $\mathcal{M}_{\mathbb{C}}$-valued random element, to a deterministic probability measure $\mu_t$ as $n\to\infty$. In particular, when $k=o(n)$, the conjectured limit is $\mu_0=\mu$, whereas for $t\neq 0$ the measures $\mu_t$ and $\mu$ should in general differ.