The phase-transition conjecture for the marginal of a random contingency table

Let $B,C>0$, let $0<\delta<1$, and let $X=(X_{ij})$ be sampled uniformly from $\mathcal M_{n,\delta}(B,C)$. Let $B_c$ denote the critical value. Phase-transition conjecture. There is a phase transition in $\delta$ for the limiting distribution of $X_{11}$ in the supercritical regime $B>B_c$: for $1/2<\delta<1$, $n^{\delta-1}X_{11}$ converges in distribution to a geometric random variable, whereas for $0<\delta<1/2$ the marginal has the normal scaling stated below. In the thin-bezel regime, the precise conjecture is

$$\frac{X_{11}-C(B-B_c)n^{1-\delta}}{n^{(1-\delta)/2}\sqrt{B_cC+(B_cC)^2}}\Longrightarrow\mathcal N(0,1).$$

The thick-bezel geometric limit is proved in the paper, while the thin-bezel normal limit and the phase transition at $\delta=1/2$ remain open.