Weak isolation conjecture for regular partial Hadamard matrices

Let $H\in M_{M\times N}(\mathbb C)$ be a dephased partial Hadamard matrix. It is regular when its row scalar products decompose as sums of cycles, isolated when it has only trivial deformations, and of Butson type when all entries are roots of unity of finite order. Weak isolation conjecture. A dephased partial Hadamard matrix $H\in M_{M\times N}(\mathbb C)$ which is regular and isolated must be of Butson type. This is proposed as a rectangular statement because the preceding square conjectures cannot both hold in that setting; the source gives no proof.