The pointed Brody principle for the Kobayashi–Royden pseudometric

Let $X$ be a compact complex manifold and let $x\in X$. The infinitesimal Kobayashi–Royden pseudometric is the pseudometric on tangent vectors defined by holomorphic disks in $X$; it is degenerate on $T_xX$ when it vanishes on a nonzero tangent vector.

Pointed Brody conjecture. If the infinitesimal Kobayashi–Royden pseudometric is degenerate on $T_xX$, then there exists a non-constant holomorphic map

$$\nf:\mathbb C\to X$$

\nwith $f(0)=x$.

This asks whether degeneracy at a prescribed point forces an entire curve passing through that point, strengthening the existence conclusion supplied by Brody's theorem, which does not ensure that the resulting curve meets the prescribed point. The source provides no resolution, so the conjecture remains open.