Generating all Ahlfors currents by a single entire curve

Let $X$ be a compact complex manifold possessing the \emph{Runge approximation property on discs}, meaning that every holomorphic map from a closed disc into $X$ is approximable by a global holomorphic map from $\mathbb{C}$. We construct an entire curve $F : \mathbb{C} \to X$ such that the associated family of concentric holomorphic discs $\{F|_{\overline{\mathbb{D}}_r}\}_{r>0}$ generates all Ahlfors currents on $X$, thereby settling a conjecture of Sibony.