The uniform effectivity conjecture for Iitaka fibrations

Let $W$ be a smooth projective variety of dimension $d$ with Kodaira dimension $\kappa(W)\ge 0$. An Iitaka fibration of $W$ is the fibration, defined up to birational equivalence, associated with the pluricanonical system and having base dimension $\kappa(W)$. Uniform effectivity conjecture. There is a natural number $m_d$ depending only on $d$ such that, for every natural number $m$ divisible by $m_d$, the pluricanonical system

$$|mK_W|$$

defines an Iitaka fibration. This asks for a dimension-dependent uniform bound on the multiples needed to realize Iitaka fibrations; the paper presents it as a question in higher dimensions, following the known surface case, while its status is not resolved in the supplied context.