Spectral variation conjecture for normal operators and differentiator compressions

Let $A$ be a normal operator on a complex Hilbert space of dimension $n\ge 2$. Let $\rho(A)$ denote the radius of the smallest closed disk containing the spectrum of $A$, let $A'$ be the compression of $A$ to the orthogonal complement of a trace vector of $A$, and let $s(A,A')$ denote the spectral variation between $A$ and $A'$. The spectral variation conjecture. One has

$$s(A,A')\le \rho(A),$$

and equality occurs if and only if either $A=0$, or $A$ has simple eigenvalues and $A^n$ is a non-zero scalar operator. The claim reformulates the polynomial conjectures in terms of spectra of a normal operator and its compression. The source gives no resolution, so the conjecture remains open there.