The uniqueness conjecture for thermal averages in two-dimensional disordered spin systems

Fix a disordered spin system of the type considered in the paper in dimension $d=2$. Let $\beta>0$ be the inverse temperature, $\tau_1,\tau_2\text{?}$ be boundary conditions in $[?]$, $[?]$ denote the box of side length $L$, and $f_v([?])$ the local observable at vertex $v$. In a translation-invariant setup, let $[?]$ denote the origin. Uniqueness conjecture. For every $\beta>0$ and disorder strength $[?]$, almost surely with respect to the disorder,

$$\lim_{L\to\infty}\sup_{\tau_1,\tau_2\in \mathcal{S}^{\mathbb{Z}^d\setminus\Lambda_L}}\frac{1}{|\Lambda_L|}\sum_{v\in\Lambda_L}\left|\left\langle f_v(\sigma)\right\rangle_{\Lambda_L}^{\tau_1}-\left\langle f_v(\sigma)\right\rangle_{\Lambda_L}^{\tau_2}\right|=0.$$

Moreover, in a translation-invariant setup,

$$\lim_{L\to\infty}\sup_{\tau_1,\tau_2\in\mathcal{S}^{\mathbb{Z}^d\setminus\Lambda_L}}\left|\left\langle f_{\mathbf 0}(\sigma)\right\rangle_{\Lambda_L}^{\tau_1}-\left\langle f_{\mathbf 0}(\sigma)\right\rangle_{\Lambda_L}^{\tau_2}\right|=0.$$

The conjecture strengthens the Aizenman--Wehr conclusion by asserting that boundary conditions cannot significantly alter thermal averages, not merely that their spatial average is insensitive to the boundary. It would imply, as a special case, the belief that the two-dimensional Edwards--Anderson spin glass has a unique ground-state pair. No resolution is supplied in the source.