The non-isolatedness conjecture for two-unitary complex Hadamard matrices

A complex Hadamard matrix is non-isolated if it belongs to a continuous family of complex Hadamard matrices rather than being an isolated point in the space of such matrices. Let $N=d^2$ and consider complex Hadamard matrices of order $N$.

Non-isolatedness conjecture. Only non-isolated complex Hadamard matrices of order $N=d^2$ provide two-unitary structures.

The claim is motivated by explicit two-unitary constructions from parametrized families, contrasted with known isolated, defect-zero matrices for which no internal parametrization is available. Whether isolated complex Hadamard matrices of square order can provide two-unitary structures remains open.