Conjectured asymptotics for the correction integrals in the inverse Kasteleyn matrix

Let $n=4m$, $0<a<1$, and take $x=(x_1,x_2)\in\mathtt{W}_{\varepsilon_1}$ and $y=(y_1,y_2)\in\mathtt{B}_{\varepsilon_2}$ with $\varepsilon_1,\varepsilon_2\in\\{0,1\\}$. Let $\zeta(x,y)$, $\Sigma(x,y)$, the contours $\mathcal{C}_j,\mathcal{C}'_k$, the contour $\gamma$, the amplitudes $A^{j,k}_{\varepsilon_1,\varepsilon_2}$, and the functions $f^-,f^+$ be as defined in the source. Correction-integral asymptotics conjecture. For $-1/\sqrt{2}\leq\alpha_x,\alpha_y<0$, $\mathcal{I}^{0,0}_{\varepsilon_1,\varepsilon_2}$ is given by the stated double-contour integral with error $O(m^{-1})$; when $\alpha_x<-1/\sqrt{2}$ or $\alpha_y<-1/\sqrt{2}$, it is given by the corresponding double-contour integral minus the $2\pi$ contour integral over $\gamma$, again with error $O(m^{-1})$. For $(j,k)\ne(0,0)$ and any $\alpha<0$, $\mathcal{I}^{j,k}_{\varepsilon_1,\varepsilon_2}$ is given by the stated double-contour integral with phase $g_{j,k}$ and error $O(m^{-1})$, where the four phases $g_{j,k}$ are the displayed combinations of $-2i(w-z)$, $\alpha_x f^\pm(w)$, and $\alpha_y f^\pm(z)$. These asymptotics are intended to provide the correction terms needed for the inverse Kasteleyn matrix and are presented without rigorous error bounds in the paper.