GPSS conjecture for geometric progress singularity-series

For integers $r\geq 4$ and $k\geq 2$, consider the cyclic Gorenstein quotient singularity of type

$$\frac{1}{\left(\frac{k^r-1}{k-1}\right)}\left(1,k,k^2,k^3,\ldots,k^{r-2},k^{r-1}\right).$$

GPSS conjecture. All such cyclic Gorenstein quotient singularities admit torus-equivariant projective, crepant, full resolutions. The theorem immediately preceding the conjecture proves the first member with $k=2$ in every dimension $r\geq 2$ and establishes uniqueness there, whereas the asserted existence for all $r\geq4$ and $k\geq2$ is presented as a conjecture and is not resolved in the supplied text.