Optimality of the lifted 8×8 bad science matrix

Determine whether the 8×8 matrix obtained by applying the lifting construction of Theorem 2 twice, starting from the optimal 2×2 matrix A2 = (1/√2) [[1, 1], [1, −1]], is optimal for the bad science matrix problem; that is, prove or refute that it maximizes β(A) = 2^{-8} ∑_{x∈{−1,1}^8} ||Ax||∞ among all 8×8 real matrices with unit ℓ2-norm rows.

Background

Using a Sylvester-style lifting, the authors construct an explicit family achieving β(A_{2^k}) = √(k+1). They prove optimality for n = 2 and n = 4 and obtain β = 2 for n = 8, matching the best known value in that dimension.

They explicitly state that whether the 8×8 instance produced by this construction is actually optimal remains open.