The Negativity Conjecture for smooth $\mathbb{Q}$-acyclic surfaces

Let $S$ be a smooth $\mathbb{Q}$-acyclic surface, and let $(X,D)$ be a log smooth completion of $S$. Here, $K_X$ is the canonical divisor of $X$, $D$ is the boundary divisor, and $\kappa$ denotes the Kodaira--Iitaka dimension. Negativity Conjecture. One has

$$\kappa\left(K_X+\frac{1}{2}D\right)=-\infty.$$

This formulation is presented as the guiding principle for studying smooth $\mathbb{Q}$-acyclic surfaces through the almost Minimal Model Program. The supplied text gives no evidence that the conjecture has been resolved.