Borcea's normal-matrix spectral-distance conjecture

Let $A$ be an $n\times n$ normal matrix. Write $W(A)$ for its numerical range, $\Sigma(A)$ for its spectrum, and let $A_k$ be the $(n-1)\times(n-1)$ matrix obtained by deleting the $k$th row and column of $A$. For compact sets, let $H$ denote the symmetrized Hausdorff distance.

Borcea's normal-matrix spectral-distance conjecture.

$$H\left(W(A),\bigcup_{k=1}^n\Sigma(A_k)\right)\leq\min_{c\in\mathbb C}\|A-cI\|.$$

This is presented as a matrix-theoretic generalization of Schmeisser's conjecture. The source provides no resolution.