The cumulant variance conjecture for the periodic directed polymer

Let $\kappa_n(t)$ be the quenched $n$-th cumulant of the endpoint of the point-to-line directed polymer, and let $R_\beta^{(\theta)}$ be the covariance function defined in the paper. Cumulant variance conjecture. For each $n\geq 0$, the asymptotic variance satisfies

$$\lim_{t\to\infty} \frac{1}{t}\operatorname{Var}[\kappa_n(t)]=(-1)^n\left.\partial_\theta^{2n}R_\beta^{(\theta)}\right|_{\theta=0}.$$

This conjecture gives the variance governing the expected central limit theorem for the centered and diffusively scaled cumulants. Its validity is motivated by a formal calculation based on the covariance identified in Theorem, but no proof or resolution is supplied here.