Complex BBM phase-diagram conjecture

Let $\beta=\sigma+i\tau\in\mathbb C$, let $\rho\in[-1,1]$, and let $p_t(\beta)$ be the log-partition function per unit time for the complex BBM energy model. Let $B_1,B_2,B_3$ denote the three regions of the complex REM phase diagram. Complex BBM phase-diagram conjecture. For every $\rho\in[-1,1]$, the complex BBM energy model has the same free energy and phase diagram as the complex REM:

$$\lim_{t\uparrow\infty}p_t(\beta)=:p(\beta)=\begin{cases}1+\frac{1}{2}(\sigma^2-\tau^2),&\beta\in\overline{B_1},\sqrt{2}|\sigma|,&\beta\in\overline{B_2},\frac{1}{2}+\sigma^2,&\beta\in\overline{B_3},\end{cases}$$

and the convergence holds in probability and in $L^1$. This conjecture asserts that correlations parametrized by $\rho$ do not change the limiting free energy or phase boundaries from those of the complex REM.