Sibony's conjecture on generating Ahlfors currents by one entire curve

Let $X$ be a compact complex manifold, and let an Ahlfors current mean a current obtained as an asymptotic limit from the concentric holomorphic discs associated to an entire curve $f\colon\mathbb{C}\to X$ along radii satisfying the Ahlfors length-area condition. Sibony's conjecture. For certain compact complex manifolds $X$, such as $\mathbb{P}^n$ or complex tori, every Ahlfors current on $X$ can be obtained from a single entire curve $f\colon\mathbb{C}\to X$. The conjecture proposes a universal entire curve generating all Ahlfors currents on these manifolds; its connection with universal entire functions and related open problems was noted by Sibony, while the statement remains open.