Appleby's conjecture on algebraic-unit normalized overlaps

Let a SIC-POVM in $\mathbb{C}^d$ have normalized overlaps $e^{i\theta_{jl}}$, where the angles $\theta_{jl}$ are defined by the normalized inner products of its fiducial vectors. An algebraic unit is an algebraic integer whose inverse is also an algebraic integer. Appleby's conjecture. The normalized overlaps $e^{i\theta_{jl}}$ are algebraic units. This conjecture concerns the arithmetic structure of SIC-POVMs and is motivated by observed connections with algebraic number theory; its general status is not resolved in the supplied source.