Airy-process limit conjecture for the rescaled Bernoulli–Exponential passage time

Let $T_h$ be the limiting passage time defined by the preceding convergence result, let $x\in\mathbb R$ be fixed, and let $\mathcal A(x)$ denote the stationary Airy process. Airy-process limit conjecture. In the parameter $x\in\mathbb R$ one has

$$\left(\frac32\right)^{4/3}h^{-5/3}\left(\frac{4h^3}{27}+\left(\frac23\right)^{5/3}h^{7/3}x-T_{h+(3/2)^{1/3}h^{1/3}x}\right)\stackrel{\mathrm d}{\Longrightarrow}\mathcal A(x)-x^2$$

as $h\to\infty$. This is the expected process-level extension of the one-point Tracy–Widom limit for the rescaled limiting passage times; the statement remains unproved in the source.