The zero-set characterization conjecture for the Kobayashi–Royden pseudometric of the spectral unit ball

Let $\Omega_n$ be the spectral unit ball, let $A\in\Omega_n$, and let $B\in\mathcal M_n$. Write $k_{\Omega_n}(A;B)$ for the Kobayashi–Royden pseudometric and $C_A$ for the cone of tangent vectors admitting a holomorphic entire disc through $A$ with derivative $B$. Zero-set characterization conjecture. One has

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

if and only if there is $\varphi\in\mathcal O(\mathbb C,\Omega_n)$ with $\varphi(0)=A$ and $\varphi'(0)=B$. In particular, if $k_{\Omega_n}(A;B)=0$, then $B\in C_A$. This conjecture seeks an exact description of the zero set of the Kobayashi–Royden pseudometric; the supplied text does not indicate whether it has been resolved.