Normal distribution and flip connectivity conjecture for random tilings of large boxes

Consider the cubiculated box

$${\cal R}=[0,L]\times[0,M]\times[0,N],\qquad L,M,N\in\mathbb{N}^*,\quad LMN\text{ even}.$$

Let $\mathbf T,\mathbf T_0,\mathbf T_1$ be independent uniformly distributed random tilings of $\mathcal R$, and let $\operatorname{Tw}$ denote the twist, while $\approx$ denotes connectivity by a finite sequence of flips. Normal distribution and flip connectivity conjecture. As $\min(L,M,N)\to\infty$, the random variable

$$\frac{1}{\sqrt{LMN}}\operatorname{Tw}(\mathbf T)$$

converges in distribution to a normal distribution centered at $0$, and

$$\lim_{\min(L,M,N)\to\infty}\operatorname{Prob}\left[\mathbf T_0\approx\mathbf T_1\mid \operatorname{Tw}(\mathbf T_0)=\operatorname{Tw}(\mathbf T_1)\right]=1.$$

The first assertion predicts Gaussian fluctuations of the twist at scale $\sqrt{LMN}$; the second says that, asymptotically, equal twist is almost surely sufficient for flip connectivity. The paper presents empirical evidence but does not establish either assertion, and notes that the formulation should hold more generally than for boxes.