Demailly's Brody–algebraic hyperbolicity conjecture

A complex projective variety $X$ is algebraically hyperbolic if there exists an $\epsilon>0$ such that every curve $C\subset X$ of geometric genus $g(C)$ satisfies

$$2g(C)-2\geq\epsilon\deg(C).$$

It is Brody hyperbolic if it admits no holomorphic maps from $\mathbb{C}$. Demailly's conjecture. A smooth complex projective variety $X$ is Brody hyperbolic if and only if it is algebraically hyperbolic. The conjecture would identify the metric and algebraic notions of hyperbolicity for smooth projective varieties; the supplied text gives no resolution status.