Universal entire-curve conjecture for Nevanlinna and Ahlfors currents

Let $X$ be a compact Oka manifold. A family of holomorphic discs generates all Nevanlinna/Ahlfors currents on $X$ if the currents associated with those discs realize every such current on $X$.

Universal entire-curve conjecture. There should be some entire curve $g:\mathbb{C}\rightarrow X$ such that the concentric discs

$$\{g\restriction_{\mathbb{D}_r}\}_{r>0}$$

generate all Nevanlinna/Ahlfors currents on $X$.

Theorem A in the source establishes a related result for compact complex manifolds satisfying the weak Oka-1 property, using discs centered at arbitrary points and of arbitrary radii. The conjecture asks for the stronger concentric-disc realization on every compact Oka manifold and is presented as an open problem.