Zauner's conjecture on Weyl–Heisenberg covariant SICs

Let $d$ be a finite dimension, let $\ket{\phi}\in\mathbb{C}^d$ be a fiducial vector for a Weyl–Heisenberg covariant SIC, and let $Z$ be the matrix with entries

$$\bra{j}Z\ket{k}=\frac{e^{i\xi}}{\sqrt{d}}\,\tau^{2jk+j^2}.$$

Zauner's conjecture. In all finite dimensions there exists a fiducial vector for a Weyl–Heisenberg covariant SIC that is an eigenvector of $Z$.