General rectangular bad science matrix problem

Determine the optimal value of β(A) = 2^{-n} ∑_{x∈{−1,1}^n} ||Ax||∞ and characterize extremal matrices for the rectangular case A ∈ ℝ^{m×n} with each row having unit ℓ2 norm, beyond the square (n×n) setting.

Background

The bad science matrix problem was originally formulated for square matrices, but its definition extends naturally to rectangular A ∈ ℝ^{m×n}. The authors emphasize that beyond the square case, essentially no results were known at the time of writing.

Although later sections address special wide cases (1×n resolved and a conjecture for 2×n), the general rectangular setting (arbitrary m and n) remains broadly open.