Zero-dimensionality conjecture for Weyl–Heisenberg-covariant SIC varieties

Let $d>3$, and consider the polynomial equations defining a SIC-POVM that is covariant under the action of the Weyl–Heisenberg group. Their solution set is an algebraic variety. Zero-dimensionality conjecture. Numerical evidence suggests that this algebraic variety has dimension zero for every $d>3$. A zero-dimensional variety has only finitely many points over an algebraic closure when the defining system is suitably interpreted; the supplied source presents this as numerical evidence rather than a proved assertion, and the general claim remains open.