Borcea's normal-matrix invertibility conjecture

Let $A$ be a normal matrix, let $\mathbf v^{(0)},\dots,\mathbf v^{(n-1)}$ be an orthonormal basis, and let $P_\ell$ denote the orthogonal projection onto $\mathbf v^{(\ell)}$. Write

$$B_\ell=P_\ell A P_\ell^*$$

for the corresponding compressions.

Borcea's normal-matrix invertibility conjecture. If every eigenvalue of every $B_\ell$ lies outside the unit disk and

$$\|(I-P_\ell)AP_\ell\|\leq1$$

for all $\ell$, then $A$ is invertible.

The statement is identified as the matrix-theoretic reduction of Statement (i), and hence as equivalent to the 2-variance conjecture in the surrounding discussion. The source provides no proof or disproof.