Conjectured optimal form for 2×n matrices

Prove that, among all 2×n real matrices A with rows normalized to unit ℓ2 norm, the matrix A = (1/√2) [[1, 1, 0, …, 0], [1, −1, 0, …, 0]] maximizes β(A) = 2^{-n} ∑_{x∈{−1,1}^n} ||Ax||∞.

Background

After solving the 1×n case via Khintchine’s inequality, the authors turn to 2×n and describe it as challenging. They present a specific candidate matrix for 2×n and state this as a conjecture.

They also note, immediately prior to the conjecture, that they believe the candidate to be correct but were unable to prove it rigorously, underscoring the open nature of this problem.