The parabolic Airy process rate-function contraction conjecture

Let $D$ be the space of admissible directed-landscape environments, let $I$ be the associated rate function, and let $Q$ be the Dirichlet energy of a function $f:\mathbb R\to\mathbb R$ defined by

$$Q(f)=\frac14\int_{\mathbb R}(f'(x)^2-4x^2)\,dx$$

for absolutely continuous $f$ satisfying $f(x)\ge -x^2$ and $f(x)-x^2\to0$ as $x\to\pm\infty$, with $Q(f)=\infty$ otherwise. The parabolic Airy process rate-function contraction conjecture. For any function $f:\mathbb R\to\mathbb R$,

$$\min\{I(e):e\in\mathcal D,\ e(0,0;\cdot,1)=f\}=Q(f).$$

This is the proposed large-deviation principle for the parabolic Airy process, obtained by analogy with the Brownian proxy and Schilder's theorem. Establishing the principle is described as an open problem and is expected to require methods handling the more delicate structure of the parabolic Airy process.