Conjecture IV'_H on orbifold Kobayashi pseudometrics

Let $c_X:X\to C(X)$ be an admissible representative of the core of $X$, and let $\Delta(c_X)$ be its base orbifold. Conjecture IV'_H. One should have

$$d_X=c_X^*\bigl(d_{(C(X)/\Delta(c_X))}\bigr).$$

In addition, if $(X/\Delta)$ is a klt orbifold of general type, its orbifold Kobayashi pseudometric should be a metric outside some proper algebraic subset $S\subset X$. This is presented as a refinement of Conjecture IV_H; the source gives no resolution.