Effective Iitaka fibration conjecture

Let $d$ be a positive integer. For a smooth projective variety $X$ with non-negative Kodaira dimension, an Iitaka fibration is a fibration birationally induced by a sufficiently divisible pluricanonical system.

Effective Iitaka fibration conjecture. There exists a positive integer $m_d$ depending only on $d$, such that for every smooth projective variety $X$ of dimension $d$ with non-negative Kodaira dimension, $|mK_X|$ defines an Iitaka fibration for every positive integer $m$ divisible by $m_d$.

The conjecture is known when $K_X$ is big, in dimensions $2$ and $3$, but remains open in dimension at least $4$.