A Conjecture on Almost Flat SIC-POVMs

A well supported conjecture states that SIC-POVMs – maximal sets of complex equiangular lines – with anti-unitary symmetry give rise to an identity expressing some of its overlaps as squares of the (rescaled) components of a suitably chosen fiducial vector. In number theoretical terms the identity essentially expresses Stark units as sums of products of pairs of square roots of Stark units. We investigate whether the identity is enough to determine these Stark units. The answer is no, but the failure might be quite mild.