Deformation conjecture for regular Hadamard matrices

A regular matrix $H\in M_N(\mathbb C)$ is a complex Hadamard matrix whose row scalar products decompose as sums of cycles, and an affine deformation is the deformation notion used in the paper. Deformation conjecture. Any regular matrix $H\in M_N(\mathbb C)$ is an affine deformation of a Butson matrix. The difficulty is to define a Butson matrix that produces $H$ through such a deformation, because decompositions into cycles and the associated parameters need not be unique. No proof is known in the source.