Regularity conjecture for Butson matrices

A partial complex Hadamard matrix $H\in M_{M\times N}(\mathbb T)$ is regular when the scalar products between its rows decompose as sums of cycles. A Butson matrix is a complex Hadamard matrix whose entries are roots of unity of finite order. Regularity conjecture. Any Butson matrix $H\in M_N(\mathbb C)$ is regular. Thus a vanishing sum of roots of unity that is not a sum of cycles cannot be used to construct a complex Hadamard matrix. The conjecture is presented as a difficult question with substantial computer evidence, but no proof is given.