Pal's permanent asymptotic conjecture for symmetric cost functions

Let $\mathcal{C}:[0,1]\times[0,1]\to\mathbb{R}$ be twice continuously differentiable up to the boundary, symmetric in its two variables, zero on the diagonal, and satisfy $\mathcal{C}(1-x,1-y)=\mathcal{C}(x,y)$. Let $\alpha,\beta:[0,1]\to\mathbb{R}$ be such that $\alpha(x)+\beta(y)$ is determined by

$$\int_0^1\exp\left(-\mathcal{C}(x,y)-\alpha(x)-\beta(y)\right)\,dy=1,$$

and

$$\int_0^1\exp\left(-\mathcal{C}(x,y)-\alpha(x)-\beta(y)\right)\,dx=1.$$

Define $\Lambda[\mathcal{C}]$ as in the preceding setup, and let $\rho(x,y)=\exp(-\mathcal{C}(x,y)-\alpha(x)-\beta(y))$. Let $I$ be the identity operator on $L^2([0,1])$, let $J$ be the rank-one operator $Jf(x)=\int_0^1f(y)\,dy$, and let $T$ be the trace-class operator

$$Tf(x)=\int_0^1\rho(x,y)f(y)\,dy.$$

Set $\mathcal{D}[\mathcal{C}]=\det_F(I+J-T^*T)$. Pal's conjecture. As $n\to\infty$,

$$\frac{1}{n!}\operatorname{perm}\Big(\big(\exp(-\mathcal{C}(i/n,j/n)):\ i,j\in[n]\big)\Big)\sim\frac{\exp(n\Lambda[\mathcal{C}])}{\sqrt{\mathcal{D}[\mathcal{C}]}}.$$

Pal motivated this conjecture by removing an additional source of randomness from earlier precise asymptotic results; the conjectured Fredholm-determinant correction is intended to capture the resulting second-order asymptotics. The paper discusses heuristic and combinatorial arguments, but the conjecture remains open in the supplied context.