Generalized Borcea variance conjecture for convex combinations of zeros

Let $F$ be a complex polynomial of degree $n\geq 2$ with zeros $z_1,\dots,z_n$ and critical points $w_1,\dots,w_{n-1}$. For $p\geq 1$, define

$$\sigma_p(F)=\min_{c\in\mathbb C}\left(\frac{1}{n}\sum_{k=1}^n|z_k-c|^p\right)^{1/p}.$$

Let $l_1,\dots,l_n$ be nonnegative weights satisfying

$$\sum_{k=1}^n l_k=1.$$

Generalized Borcea variance conjecture. For every $p\geq 1$ and every such choice of weights,

$$\min_{1\leq j\leq n-1}\left|\sum_{k=1}^n l_kz_k-w_j\right|\leq \sigma_p(F).$$

This extends Borcea's variance conjecture in the same way that Schmeisser's conjecture extends Sendov's conjecture; the paper does not report a resolution.