Zauner's strong conjecture on Weyl–Heisenberg SIC POVMs

Let $d\geq 2$, let $\mathbb{Z}_d=\{0,1,\ldots,d-1\}$, and let $W_{i,j}=U^iV^j$ be the discrete Weyl matrices on $\mathbb{C}^d$, with $U$ the cyclic shift and $V$ the diagonal matrix of $d$-th roots of unity. A unit vector $w\in\mathbb{C}^d$ is a fiducial vector when its Weyl orbit generates a SIC POVM. Zauner's strong conjecture. For every positive integer $d\geq 2$, there exists a unit vector $w\in\mathbb{C}^d$ such that

$$\{W_{i,j}w:(i,j)\in\mathbb{Z}_d\times\mathbb{Z}_d\}\subseteq\mathbb{C}^d$$

generates a SIC POVM. It is a stronger, group-covariant form of Zauner's weak conjecture, and the source notes that it is not known whether the weak conjecture implies this stronger conjecture; its general status is open.