Law of large numbers for roots of finite free multiplicative convolution

Let $p_d$ be a monic polynomial of degree $d$ with non-negative real roots

$$\Lambda_d=\{\lambda_{1,d}\geq\dots\geq\lambda_{d,d}\}.$$

Let $\mu\in\mathcal{P}_{+,c}$, and suppose that the empirical root distributions of $p_d$ converge weakly to $\mu$:

$$\frac{1}{d}\sum_{i=1}^{d}\delta_{\lambda_{i,d}}\xrightarrow{w}\mu$$

as $d\to\infty$. Law of large numbers for roots. Then

$$\frac{1}{d}\sum_{i=1}^{d}\delta_{R_i(\Lambda_d)}\xrightarrow{w}\Phi(\mu)$$

as $d\to\infty$.

This conjecture predicts that the limiting empirical distribution of the roots produced by finite free multiplicative convolution is determined by the limiting empirical root distribution through the map $\Phi$, extending the examples discussed in the paper. The supplied text does not state whether the conjecture has been resolved.