The extended-kernel limit for the pentagonal Aztec diamond

Let $X_N^R$ denote the top line of an Aztec diamond of size $N$ with a triangle removed from the top corner 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 given observation times $T_1,\ldots,T_k$ and thresholds $U_1,\ldots,U_k$, the pentagonal Aztec-diamond conjecture.

$$\lim_{N\to\infty}\mathbf P\left(\bigcap_{\ell=1}^k\{X_N^{R,\mathrm{resc}}(T_\ell)\leq U_\ell\}\right)=\det\left(\mathbbm{1}-\widehat K^{\mathrm{ext}}\right)_{L^2(E)},$$

where

$$E=\{(T_1,[U_1-R,0])\times\ldots\times(T_k,[U_k-R,0])\}.$$

This predicts an extended-kernel determinantal scaling limit for the top line when the triangular cutout becomes, after rescaling, a conditioning to remain below the fixed height $R$.