The pathwise variational principle for skew Young tableaux

Let $\lambda_N/\lambda'_N$ be a sequence of skew Young diagrams with $n_N=|\lambda_N/\lambda'_N|$ cells. Let $g_0(t,x)$ be any other continuous and almost everywhere smooth function satisfying the same boundary conditions as the asymptotic shape and

$$\forall t\in[0,1],\qquad \int_{\mathbb{R}}\bigl(g_0(t,x)-g_0(0,x)\bigr)\,dx=t.$$

Let $F_{\varepsilon,g_0}^{\lambda_N/\lambda'_N}$ denote the number of tableaux whose corresponding function is $\varepsilon$-close to $g_0$ in the $C^0$ topology.

Pathwise asymptotic enumeration conjecture. As $n_N\to\infty$ and $\varepsilon\to0$,

$$\log F_{\varepsilon,g_0}^{\lambda_N/\lambda'_N}=\frac{1}{2}n_N\log n_N+n_N\mathcal{L}[g_0]+o(n_N),$$

with the error understood through

$$\lim_{\varepsilon\to0}\limsup_{N\to\infty}\frac{1}{n_N}\left(\log F_{\varepsilon,g_0}^{\lambda_N/\lambda'_N}-\frac{1}{2}n_N\log n_N+n_N\mathcal{L}[g_0]\right)=0.$$

This conjecture would extend the variational description from the maximizing limit shape to the asymptotic number of tableaux following any admissible macroscopic trajectory. It is presented as the reason the functional appears in the preceding conjecture; no resolution is supplied in the source.