Conjecture on small complex Hadamard matrices with noncommutative entries

A complex Hadamard matrix is a square matrix whose entries are unitary elements of a unital $C^*$-algebra and whose distinct rows and columns are orthogonal. In the size-four case, Proposition 1.8 gives a family of such matrices; an entry is self-adjoint when it equals its adjoint.

Small-size classification conjecture. The matrices in Proposition 1.8 are the only $4\times4$ Hadamard matrices, and there is no $5\times5$ Hadamard matrix with self-adjoint entries.

This proposes a classification in size four and rules out the specified self-adjoint examples in size five. The supplied text gives no resolution status for either assertion.