Conjecture IV_H on the Kobayashi metric of the core

Let $X\in\mathcal C$ be a compact complex manifold with core $c_X:X\to C(X)$, and let $d_{C(X)}^*$ be the induced pseudometric on the core. Conjecture IV_H. The pseudometric $d_{C(X)}^*$ is a metric on some Zariski nonempty open subset $C(X)^*\subset C(X)$. The source gives no resolution.