Minimum dimension for strictness of the ξ_n–θ_n inequality on the cube

Prove or refute that the minimum integer n for which the inequality ξ_n(Q_n) < ((n+1)/2)(θ_n(Q_n) − 1) + 1 holds is n = 4, where Q_n = [0,1]^n and ξ_n(Q_n) denotes the minimal absorption index of Q_n by an inner nondegenerate simplex while θ_n(Q_n) denotes the minimal operator norm of first-order Lagrange interpolation projectors on Q_n.

Background

The inequality ξ_n(Q_n) ≤ ((n+1)/2)(θ_n(Q_n) − 1) + 1 holds for all n. Equality is known for n = 1, 2, 3, and 7, where both ξ_n(Q_n) and θ_n(Q_n) have been exactly determined.

The conjecture proposes that the first dimension where the inequality becomes strict is n = 4, but this has not been proven.