Polynomial realization conjecture for zero Kobayashi–Royden metric in the spectral ball

Let $obreak\Omega_n$ be the spectral ball of $n\times n$ complex matrices, let $A\in\Omega_n$ and $B\in\mathcal M_n$, and let $obreak\kappa_{\Omega_n}(A;B)$ denote the Kobayashi–Royden metric at $A$ in the direction $B$. Write $sp(A)$ for the spectrum of $A$.

Polynomial realization conjecture. If

$$\kappa_{\Omega_n}(A;B)=0,$$

then there is a polynomial mapping $p:\mathbb C\to\Omega_n$ of degree at most $n$ such that

$$p(0)=A,\qquad p'(0)=B,\qquad sp(p(\lambda))=sp(A)\quad\text{for every }\lambda\in\mathbb C.$$

The paper proves the conjecture for $n=2$; the general case is left open. The statement strengthens the known existence of an entire mapping with the same initial data and constant spectrum by requiring a polynomial of degree at most $n$.