Asymptotic midpoint uniform convexity and unbounded diamond distortion in a reflexive tree space

We construct a separable reflexive real Banach space whose given norm is asymptotically midpoint uniformly convex but which admits no asymptotically uniformly convex equivalent norm. Its averaged midpoint modulus is at least $\sqrt{1+t^2/12}-1$, and the countably branching diamond of depth k has distortion at least $\sqrt{1+k/12}$ in this space. This gives a negative answer to the reflexive diamond converse for asymptotic uniform convexifiability.