Optimality of the presented 5×5 bad science matrix

Determine whether the explicit 5×5 matrix A = (1/(2√3)) · [[2, 2, 0, 0, 2], [−2, 2, 0, 2, 0], [−2, 0, 0, −2, 2], [0, −√3, √3, √3, √3], [0, √3, √3, −√3, −√3]] maximizes β(A) = 2^{-5} ∑_{x∈{−1,1}^5} ||Ax||∞ over all 5×5 real matrices whose rows have unit ℓ2 norm.

Background

The authors present what is currently the best known 5×5 example and describe its image structure on the hypercube vertices (eight mapped to max-entry 2, the remaining twenty-four to max-entry √3). Despite this strong candidate, they explicitly acknowledge that its optimality is unknown.

Later in the paper they resolve optimality for n ≤ 4, but the n = 5 case remains unsettled, making the optimality of this specific 5×5 construction a concrete open question.