Converse inequality for r_{S*}(ε) in the dual of Schlumprecht’s space

Determine whether, for every ε in (0,2), the optimal radius r_{S*}(ε) of the largest ball centered at the origin contained in the ε-Szlenk derivation s_ε B_{S*} of the dual unit ball of the dual of Schlumprecht’s space S satisfies the lower bound r_{S*}(ε) ≥ inf_{n∈N} ((log2(n+2) − ε) / log2(n+1)).

Background

Section 3 studies Szlenk derivations for Tsirelson’s space and the dual of Schlumprecht’s space S*. The authors define r_X(ε) as the largest radius r > 0 such that the ball r B_{X*} is contained in the ε-Szlenk derivation s_ε B_{X*} of the dual unit ball. They establish the upper bound (6): r_{S*}(ε) ≤ inf_{n∈N} ((log2(n+2) − ε) / log2(n+1)) for all ε ∈ (0,2).

The open problem asks whether the reverse inequality holds, which would identify the exact value of r_{S*}(ε). Establishing the converse would resolve whether the upper bound (6) is sharp and fully characterize the largest ball contained in s_ε B_{S*} for the dual of Schlumprecht’s space.