Conditional law of large numbers for directed last passage percolation

Let $a,b>0$ and let $\frac{1}{\bar{\rho}}$ be the lower and $\bar{\rho}$ the upper slope determined by the conditioning parameters, with $\bar{\rho}=\frac{\bar{\rho}_{\rm num}}{\bar{\rho}_{\rm den}}$ as in the source. For $(x,y)\frac{}{ }\in\mathbb R_+^2$, define

$$\Omega_1=\{(x,y)\in(1,\infty)^2: \frac{1}{\bar{\rho}}<\frac{y-1}{x-1}<\bar{\rho}\},\qquad \Omega_2=\{(x,y)\in\mathbb R_+^2: \frac{1}{\bar{\rho}}<\frac yx<\bar{\rho}\}.$$

Under the assumptions of the conditional law of large numbers theorem, let $h$ denote the limiting shape function and condition on $\mathcal L(aN,bN)=\ell N$.

Conditional law of large numbers conjecture. The convergence in the conditional law of large numbers holds with

$$h(x,y)=\begin{cases} \ell+\bar{\mathcal L}_{(a,b)}(xa,yb),&(x,y)\in\Omega_1,\\ \frac12\left[(\ell+a-b)x+(\ell-a+b)y-|x-y|\sqrt D\right],&(x,y)\in\Omega_2\setminus\Omega_1,\\ \bar{\mathcal L}(xa,yb),&(x,y)\in\mathbb R_+^2\setminus\Omega_2. \end{cases}$$

This conjecture describes the macroscopic conditional profile throughout the positive quadrant: inside the high-time region the profile contains the conditioned value, in the intermediate region it is linear, and outside it has the usual limiting shape. Its status is unresolved in the supplied source.