The Weak Specialness Conjecture for entire curves and function fields

Let $X$ be a projective variety defined over ${\mathbb C}$. An entire curve on $X$ is a holomorphic map from ${\mathbb C}$ to $X$. Let $Y_{\mathrm{exc}}$ be a proper closed subset, let $\pi:X\to Y$ be a dominant map, let $\mathcal L$ be an ample line bundle on $Y$, and let $\alpha$ be a positive constant. Weak Specialness Conjecture. The variety $X$ is weakly special if and only if both of the following conditions hold: there exists an entire curve ${\mathbb C}\to X$ with Zariski-dense image; and there are no dominant map $\pi:X\to Y$, ample line bundle $\mathcal L$, positive constant $\alpha$, and proper closed subset $Y_{\mathrm{exc}}$ such that, for every smooth integral curve ${\mathcal C}$ and morphism $s:{\mathcal C}\to X$ with $s({\mathcal C})$ not contained in $Y_{\mathrm{exc}}$,

$$\deg (\pi\circ s)^*\mathcal L\leq\alpha\left(2g({\mathcal C})-2\right).$$

This is presented as the analogue of the rational-point conjecture for entire curves and function fields. The supplied text does not state whether it is open or resolved.