The Airy process limit for the L-shaped Aztec diamond

Let $X_N^R$ denote the top line of an Aztec diamond of size $N$ with a square removed from the top so that its lower tip is at height $N/\sqrt{2}+2^{-5/6}N^{1/3}R$. Define

$$X_N^{R,\mathrm{resc}}(T)=\frac{X_N^R(2^{-1/6}N^{2/3}T)-N/\sqrt{2}}{2^{-5/6}N^{1/3}}.$$

For any $T_1<T_2<\ldots<T_k$ and $U_1,\ldots,U_k\leq R$, the L-shaped Aztec-diamond conjecture.

$$\lim_{N\to\infty}\mathbf P\left(\bigcap_{\ell=1}^k\{X_N^{R,\mathrm{resc}}(T_\ell)\leq U_\ell\}\right)=\frac{\mathbf P\left(\bigcap_{\ell=1}^k\{{\mathcal A}_2(T_\ell)-T_\ell^2\leq U_\ell\}\cap\{{\mathcal A}_2(0)\leq R\}\right)}{\mathbf P({\mathcal A}_2(0)\leq R)},$$

where ${\mathcal A}_2$ is the Airy$_2$ process. Consequently,

$$\lim_{N\to\infty}\mathbf P(X_N^{R,\mathrm{resc}}\leq U)=\frac{F_{\mathrm{GUE}}(\min\{U,R\})}{F_{\mathrm{GUE}}(R)},$$

where $F_{\mathrm{GUE}}$ is the GUE Tracy--Widom distribution function. This predicts that the rescaled top line in the L-shaped geometry converges to the Airy$_2$ process with parabolic shift, conditioned by the event ${\mathcal A}_2(0)\leq R$; the stated limit is expected from the corresponding Aztec-diamond scaling limit.