Fourier characterization of stationary Hadamard matrix models

Let a Hadamard matrix model be a matrix model arising from a complex Hadamard matrix, and call it stationary on its image when its associated integration state has the stationarity property on the image quantum group. Fourier matrix models are the models arising from Fourier-type Hadamard matrices. Fourier stationarity conjecture. The only Hadamard matrix models which are stationary on their images are the Fourier matrix models. This is motivated by the similarities between Hadamard and Weyl matrix models and by the stationary examples discussed in the cited work; the claimed converse is left open in the supplied text.