Establish the unlocked-phase cross-ratio conjecture

Establish the thermodynamic identity $\mathcal R_\alpha^{(\times)}=2^{\alpha-1}$ for the normalization-free free/fixed cross-ratio throughout the unlocked phase $0<\alpha<4$.

Background

The second normalization-free ratio is independent of the character central-limit hypothesis controlling $\mathcal R_\alpha^{(\eta)}$. Exact values are known at $\alpha=1$ and $\alpha=2$, while numerical and Pfaffian data support the proposed formula at additional indices.

The conjecture would imply an Rényi-index-independent entropy combination $\Delta_\alpha^{(\times)}=-\log 2$ throughout the unlocked phase. The paper explicitly says that this behavior, especially near $\alpha=4$, lacks an analytic proof.