Hyperbolicity conjecture for general hypersurfaces of degree at least $2n-1$

Let $X\subset \mathbb{P}^{n}$ be a general hypersurface with $n\geq 3$ and degree $\deg X\geq 2n-1$. The hypersurface $X$ is hyperbolic, meaning that its Kobayashi pseudodistance is a distance.

Hyperbolicity conjecture for general hypersurfaces. A general hypersurface

$$X\subset \mathbb{P}^{n},\qquad n\geq 3,\qquad \deg X\geq 2n-1,$$

is hyperbolic.

This is the compact counterpart of the logarithmic hyperbolicity question for the complement of a hypersurface. The source states the precise degree threshold as a conjectural assertion and supplies no evidence of resolution.