The hierarchy invariant conjecture for links of terminal quotient singularities

Let $G\subset U(n)$ be a discrete subgroup such that $\mathbb{C}^n/G$ is an isolated singularity, and let $\operatorname{H_{cx}}$ be the hierarchy invariant with its semi-dilation grading. The link is $S^{2n-1}/G$ with the standard contact structure $\xi_{std}$.

The terminal-singularity conjecture. If the singularity $\mathbb{C}^n/G$ is terminal, then

$$\operatorname{H_{cx}}(S^{2n-1}/G,\xi_{std})=0^{\operatorname{SD}}.$$

The claim seeks a contact-topological consequence of the algebro-geometric condition of terminality for quotient singularities, via the hierarchy functor. Its resolution is not supplied in the paper.