The variational principle and asymptotic enumeration for skew Young tableaux

Let $\lambda_N/\lambda'_N$ be a sequence of skew Young diagrams with $n_N=|\lambda_N/\lambda'_N|$ cells and limiting rotated, rescaled shape $L/L'$. Let $g(t,x)$ be the asymptotic shape of a random skew Young tableau, with boundary values $g(0,x)$ and $g(1,x)$ given by $L$ and $L'$, respectively. For a function $g$, write $g'_t$ and $g'_x$ for its partial derivatives.

The variational principle for skew Young tableaux. The function $g(t,x)$ maximizes

$$\mathcal{L}[g]=\int_0^1\int_{\mathbb{R}}g'_t\left(-\log g'_t+\log\cos\frac{\pi g'_x}{2}\right)\,dx\,dt-\log\frac{\pi}{\sqrt{2}}.$$

Moreover,

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

where $F^{\lambda_N/\lambda'_N}$ is the number of tableaux and $g$ is scaled to have area $1$.

This conjecture gives an explicit variational functional for the limit shape and the exponential asymptotics of the number of skew Young tableaux. The preceding work cited in the source established the existence and uniqueness of a maximizing function, while the explicit form and the stated enumeration asymptotics are conjectured here.