Real-real symmetric-state conjecture for Fourier observables

Let $\rho$ be an optimal state in the Fourier case. A real-real symmetric state is a state of the form specified in the source by equation $\mathrm{(rrsState)}$.

Real-real symmetric-state conjecture. There exists a real-real symmetric state $\sigma$ with the same entropy pair as $\rho$.

If true, this would restrict the search for optimal states in the Fourier case to real-real symmetric states, substantially reducing the problem's complexity.