Finite free convolution moment comparison conjecture

Let $p$ and $q$ be real-rooted polynomials of degree $d$, and let $\f$ be in $C^{4}(\mathbb{R})$ with $f^{(4)}\geq 0$. Write $\mu_p$ and $\mu_q$ for the probability measures assigning equal point masses to the roots of $p$ and $q$, respectively, and let $p\boxplus_d q$ denote their finite free additive convolution. Finite free convolution comparison conjecture. Then

$$\frac{1}{d}\sum_{x:(p\boxplus_d q)(x)=0}f(x)\leq\int f(t)\,\operatorname{d}(\mu_p\boxplus\mu_q)(t).$$

This conjecture extends the known comparison of the supports of finite free and free convolution. The source states that it is true up to leading order and that the condition $f^{(4)}\geq 0$ is necessary, but the full assertion remains open.