Conjecture III_H: special manifolds and Kobayashi speciality

Let $X\in\mathcal C$, and call $X$ $H$-special when its Kobayashi pseudometric satisfies $d_X\equiv 0$. Conjecture III_H. The manifold $X$ is special if and only if it is $H$-special. The source presents this as a conjectural characterization of special manifolds by vanishing Kobayashi pseudometric; no resolution is given.