The trichotomy conjecture for cyclic dilations of affine varieties

Let $M$ be an $n$-dimensional smooth affine variety over a field $\mathbb{K}$ of characteristic zero. Let $\kappa(M)$ denote its relevant Kodaira dimension. A cyclic dilation is understood in the sense of the source, and a quasi-dilation is likewise understood in the source; $h$ denotes the associated parameter. Trichotomy conjecture. The following hold:

• If $\kappa(M)=-\infty$, then $M$ admits a cyclic dilation with $h=1$.
• If $\kappa(M)=0$, then $M$ admits a cyclic dilation if and only if it admits a quasi-dilation with $h\neq1$.
• If $\kappa(M)=n$, then $M$ never admits a cyclic dilation.

The conjecture proposes a classification in characteristic zero according to the Kodaira dimension. The source notes that manifolds with vanishing symplectic cohomology must be regarded as carrying cyclic dilations for the statement to make sense, and mentions McLean's conjecture that these are precisely the manifolds ruled by affine lines; no resolution of the trichotomy is given.