The equivalence conjecture for analytic specialness

Let $(X,\Delta)$ be a smooth proper orbifold pair. Call it Kobayashi-special when its Kobayashi pseudometric vanishes identically, Picard-special when it admits the specified dense-degeneration orbifold map from the punctured disk, Borel-special when it admits a Zariski-dense graph from a smooth affine curve, and Brody-special when it admits a holomorphic orbifold map from $\mathbb{C}$ with Zariski-dense image. Analytic specialness conjecture. The following are equivalent: $(X,\Delta)$ is special, Kobayashi-special, Picard-special, Borel-special, or Brody-special. The conjecture extends a collection of conjectures on special varieties; the source gives no resolution.