Flat inverse Steklov spectral conjecture

Let two flat surfaces be equipped with their Steklov spectra. Flat inverse Steklov spectral conjecture. Two flat surfaces are Steklov isospectral if and only if they are isometric. In the flat setting, conformally equivalent metrics whose conformal factor equals one on the boundary coincide, so the general $\sigma$-isometry conjecture reduces to isometry. The supplied text does not state whether this flat version has been resolved.