  We examine universal positivity constraints on $2 \to 2$ scattering in 4d
planar $N=4$ supersymmetric Yang-Mills theory with higher-derivative
corrections. We present numerical evidence that the convex region of allowed
Wilson coefficients (the ``EFT-hedron'') flattens completely along about
one-third of its dimensions when an increasing number of constraints on the
spectral density from crossing-symmetry are included. Our analysis relies on
the formulation of the positivity constraints as a linear optimization problem,
which we implement using two numerical solvers, SDPB and CPLEX. Motivated by
the flattening, we propose a novel partially resummed low-energy expansion of
the $2 \to 2$ amplitude. As part of the analysis, we provide additional
evidence in favor of the conjecture [1] that the Veneziano amplitude is the
only amplitude compatible with both S-matrix bootstrap constraints and string
monodromy.
