Schmeisser's weighted generalization of Sendov's conjecture

Let $F$ be a complex polynomial of degree $n\geq 2$ with zeros $z_1,\dots,z_n$, all satisfying $|z_k|\leq 1$, and critical points $w_1,\dots,w_{n-1}$. Let $l_1,\dots,l_n$ be nonnegative weights satisfying

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

Schmeisser's conjecture. For every such choice of weights,

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

This generalizes Sendov's conjecture by replacing an individual zero with an arbitrary convex combination of the zeros; the paper gives no resolution.