The Gutt–Hutchings capacity conjecture for affine varieties of negative Kodaira dimension

Let $M$ be a smooth affine variety with $\kappa(M)=-\infty$, and let $c_1^{\mathit{GH}}(M)$ denote its first Gutt–Hutchings capacity. Gutt–Hutchings capacity conjecture.

$$c_1^{\mathit{GH}}(M)<\infty.$$

The source presents this as a statement implying the first item of the trichotomy conjecture, since finiteness of $c_1^{\mathit{GH}}(M)$ is equivalent there to the existence of a cyclic dilation with $h=1$; no resolution is stated.