Hyperbolicity conjecture for adjoint linear series

Let $X$ be a smooth projective variety of dimension $n\geq 3$, and let $L$ be an ample line bundle on $X$. The linear system $|K_X+mL|$ is the adjoint linear series associated to $L$.

Hyperbolicity conjecture. The linear system $|K_X+mL|$ is hyperbolic if $m\geq 3n+1$.

This is motivated by Fujita-type conjectures and the expectation that sufficiently positive linear systems on smooth projective varieties exhibit hyperbolicity. The supplied source does not establish that the assertion is resolved.