Existence of dextrodual operator-orbit frames beyond the weighted auxiliary sequence form

Determine whether, for finite singular Borel measures μ on [0,1), there exist frames {T^n g0}_{n≥0} in L^2(μ) that are dextrodual to {e^{2π i n x}}_{n≥0} but are not representable in the form g_n = g0 h_n, where h_n is the Kaczmarz auxiliary sequence of {e^{2π i n x}} in L^2(g0 μ) as specified in Proposition ‘exist’.

Background

Proposition ‘exist’ provides a constructive family of frames {T^n g0} dextrodual to the exponentials for singular measures by taking g_n = g0 h_n, with h_n the auxiliary Kaczmarz sequence in L^2(g0 μ).

The authors raise the question of whether all such dextrodual operator-orbit frames must be of this form or if genuinely different constructions exist, noting that they were unable to resolve this existence question.