Five-moment conjecture for GUE fluctuations of directed random polymers

Let $\omega$ be an i.i.d. random environment. For each $\beta>0$, let $Z_n^{\omega}(0;\beta)$ be the directed polymer partition function, and let $F_{GUE}$ denote the Tracy–Widom GUE distribution. Five-moment conjecture. If $\omega$ has $5$ moments, then there are constants $c=c(\beta)$ and $\sigma=\sigma(\beta)$ such that, for all $\beta>0$ and as $n\to\infty$,

$$\frac{\log Z_n^{\omega}(0;\beta)-c(\beta)n}{\sigma(\beta)n^{1/3}}\xrightarrow{(d)}F_{GUE}.$$

This is presented as a conjectural strong-universality statement for directed polymers, contrasting with the paper's weaker result under a small-$\beta$ exponential-moment assumption. Its resolution is not given in the supplied text.