Limit law for repeated derivatives of polynomials with i.i.d. roots

Let $P_n(z)=\prod_{k=1}^n(z-Z_k)$, where $Z_1,\ldots,Z_n$ are i.i.d. random variables with rotationally invariant law $\mu_0$ on $\mathbb{C}$ having no atom at $0$. Let $\mu_t$ be the deterministic rotationally invariant measure defined by the formulas referenced in the source. i.i.d.-root limit conjecture. For every $t\in[0,1)$,

$$\frac1n\sum_{z\in\mathbb{C}:P_n^{([tn])}(z)=0}\delta_z\overset{P}{\underset{n\to\infty}{\longrightarrow}}\mu_t$$

in probability on $\mathcal M(\mathbb{C})$. The conjecture predicts that the random-root model has the same asymptotic repeated-derivative behavior as the deterministic-coefficient model constructed from the corresponding convex function; the claim is open.