Symplectic Zauner's conjecture for real symplectic equiangular tight frames

Let $d$ be the dimension of a real symplectic vector space, and let a $d\times n$ ETF denote an equiangular tight frame in that space. Symplectic Zauner's conjecture. There exists a $d\times n$ ETF in real symplectic space if and only if

$$n = \begin{cases} d & \text{if }d\equiv 0 \text{ mod }4 \text{ or } d = 2, \\ d+1 & \text{if }d\equiv 2 \text{ mod }4. \end{cases}$$

The preceding theorem proves the necessary direction, so the conjecture reduces to existence in the listed cases. It is presented as the symplectic analogue of Zauner's conjecture; the supplied text does not report whether the remaining sufficiency direction is known.