Conditional fluctuation conjecture for directed last passage percolation

Let $\Omega_1$ and $\Omega_2$ be the regions from the conditional law of large numbers conjecture, and let $h(x,y)$ be its limiting profile. Under the assumptions and notation of the diagonal and same-side off-diagonal fluctuation theorems, condition on $\mathcal L(aN,bN)=\ell N$.

Conditional fluctuation conjecture. The following hold. For $(x,y)\in\Omega_1$, the centered passage time has Tracy–Widom $\operatorname{TW}_2$ fluctuations on the $N^{1/3}$ scale with normalization

$$(ab(x-1)(y-1))^{-1/6}(\sqrt{a(x-1)}+\sqrt{b(y-1)})^{4/3}N^{1/3}.$$

For $(x,y)\in\Omega_2\setminus\overline{\Omega}_1$, the $N^{1/2}$-scaled fluctuations converge in finite-dimensional distributions to the stated Brownian-bridge processes, with independent standard Brownian bridges on the two sides of the diagonal. For $(x,y)\in\mathbb R_+^2\setminus\overline{\Omega}_2$, the centered passage time has $\operatorname{TW}_2$ fluctuations on the $N^{1/3}$ scale with normalization

$$(abxy)^{-1/6}(\sqrt{ax}+\sqrt{by})^{4/3}N^{1/3}.$$

The conjecture extends the proved diagonal and off-diagonal fluctuation results to the full conditional phase diagram. The source explicitly marks the middle assertion as an expectation and provides a heuristic argument; no resolution is supplied.