Conjecture on the non-integer BP solution for the permanent

Let $P$ be a nonnegative matrix, let $T_c$ be the critical temperature from Proposition 1, and let $\beta_i^j$ denote a solution of the belief-propagation (BP) equations on the edge set $E$. The solution under consideration is the non-integer BP solution that emerges from the maximum-likelihood configuration at $T=T_c$.

Non-integer BP solution conjecture. The non-integer solution extends beyond the small $T_c+\varepsilon$ vicinity of $T_c$ and transitions smoothly as $T\to\infty$ into the fully homogeneous solution

$$\beta_i^j=1/N\quad\text{for all }(i,j)\in E.$$

Moreover, no other non-integer solutions exist for $T<T_c$; consequently, when this solution emerges at $T=T_c$, it gives a global minimum of the belief-propagation free energy.

This conjecture concerns the global continuation and uniqueness of the non-integer BP solution beyond the local analysis near the critical temperature. The source presents these claims as plausible conjectures and does not provide a resolution.