Conjectured mesoscopic asymptotics for the inverse Kasteleyn matrix

Let $x \in \mathtt{W}_{\varepsilon_1}$ and $y \in \mathtt{B}_{\varepsilon_2}$ with $\varepsilon_1,\varepsilon_2 \in \\{0,1\\}$. Let $K_0(\alpha)$ and $K_1(\alpha)$ denote modified Bessel functions of the second kind, and let $\zeta(x,y)$, $\Sigma(x,y)$, and the asymptotic coordinates $\alpha_x,\alpha_y$ be as defined in the source. Let $I_0,I_1,I_2,I_3,I_4$ and $\psi$ be the integrals and combination specified there. The inverse-Kasteleyn asymptotics conjecture. In the limit $m\to\infty$ with $a=1-Bm^{-1/2}$ and $\alpha_x\ne\alpha_y$, the entries $K_a^{-1}(x,y)$ satisfy the two displayed asymptotic formulas in the source: for $-1/\sqrt{2}\leq\alpha_x,\alpha_y<0$, the formula involving $K_0$, $K_1$, and $\psi$, and for the stated region with a coordinate below $-1/\sqrt{2}$, the formula involving $K_0$, $K_1$, $I_0$, and $\psi$, with error $o(m^{-1/2})$. These formulas describe the two-point correlation kernel in the mesoscopic limit of the two-periodic weighted Aztec diamond. They are conjectural because the paper does not provide a proof of the stated asymptotics.