Universality of directed polymers in the intermediate disorder regime

In 2018, Krishnan and Quastel showed that the fluctuations of Seppäläinen’s log-gamma polymer converge in law to the Tracy–Widom GUE distribution in the intermediate disorder regime, which corresponds to taking the inverse temperature $\beta$ to depend on the length of the polymer $2n$, with $\beta =n^{-\alpha }$ for some $\alpha <1\mathrm{/}4$. They also conjectured that this should hold for directed polymers with general i.i.d weights. We prove that their conjecture is true for any $1\mathrm{/}5<\alpha <1\mathrm{/}4$.