Minimal-support variance conjecture for planar point-mass measures

Let $\mu$ be a finite point-mass probability measure on $\mathbb C$. Let $S(\mu)$ be its finite support, let $S_{\min}(\mu)$ consist of support points having minimal mass, and let $V(\mu)$ be the set of zeros of its Cauchy transform outside $S(\mu)$. Let $h$ denote the one-sided Hausdorff distance and let $\sigma_\infty(\mu)$ be the Chebyshev radius of $S(\mu)$.

Minimal-support variance conjecture.

$$h(S_{\min}(\mu),V(\mu))\leq\sigma_\infty(\mu).$$

The source notes that a related stronger claim fails for three-point measures and presents this restricted statement as one that still implies Sendov's conjecture. Its resolution is not given.