Must a reflexive Banach space with no AUC renorming contain the countably branching diamonds uniformly? (Baudier-Lancien Problem 39)

Theorem 1.1: the dual $X=J^*$ of a countably branching segment-forest space with finite, unbounded component heights is infinite-dimensional, separable and reflexive, admits no equivalent AUC norm, and its given norm has averaged midpoint modulus $\widehat\delta_X(t)\ge\sqrt{1+t^2/12}-1$; every embedding of $D_k$ into $X$ has distortion at least $\sqrt{1+k/12}$. So Problem 39 has a negative answer, and AMUC does not imply AUC renormability even for reflexive spaces (Baudier had shown the non-reflexive separation in 2026). Companions add midpoint-lens estimates, a sharper $D^2\ge1+k/4$ bound, exact moduli in a Daugavet subspace of $L_1$, and further tree-potential and $L_1$ examples. Not shown: sharp distortion growth rates.

Theorem 1.1: the dual $X=J^*$ of a countably branching segment-forest space with finite, unbounded component heights is infinite-dimensional, separable and reflexive, admits no equivalent AUC norm, and its given norm has averaged midpoint modulus $\widehat\delta_X(t)\ge\sqrt{1+t^2/12}-1$; every embedding of $D_k$ into $X$ has distortion at least $\sqrt{1+k/12}$. So Problem 39 has a negative answer, and AMUC does not imply AUC renormability even for reflexive spaces (Baudier had shown the non-reflexive separation in 2026). Companions add midpoint-lens estimates, a sharper $D^2\ge1+k/4$ bound, exact moduli in a Daugavet subspace of $L_1$, and further tree-potential and $L_1$ examples. Not shown: sharp distortion growth rates.