Zauner–Appleby conjecture on Weyl–Heisenberg SIC fiducials

Let $U_{\mathcal{Z}}$ be the Zauner unitary associated with the Zauner symplectic matrix

$$\mathcal{Z}=\begin{pmatrix}0&-1\\1&-1\end{pmatrix}.$$

A Weyl–Heisenberg SIC fiducial is a fiducial vector generating a symmetric informationally complete measurement under Weyl–Heisenberg translations. Zauner–Appleby conjecture. In every $d$-dimensional Hilbert space, there exists a Weyl–Heisenberg SIC fiducial that is an eigenvector in the largest eigenspace of $U_{\mathcal{Z}}$. This symmetry prediction is supported by all dimensions in which exhaustive searches have been performed, but remains open in general.