Conjecture on strict Valiant-rigidity of known Hadamard matrices

A Hadamard matrix is a square matrix whose entries are $+1$ and $-1$ and whose rows are mutually orthogonal. A matrix is strictly Valiant-rigid when its strict rigidity does not admit the sublinear upper bounds that define non-rigidity over its field of definition. Babai's conjecture. The family of known Hadamard matrices is not strictly Valiant-rigid. This conjecture is motivated by the fact that Walsh–Hadamard matrices and Paley–Hadamard matrices have already been shown not to be strictly or absolutely Valiant-rigid, respectively; it concerns whether the same phenomenon holds for the broader family of known Hadamard matrices.