  In 2018, Krishnan and Quastel showed that the fluctuations of
Sepp\"al\"ainen'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 $\alpha<1/4$. 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 $1/5<\alpha<1/4$.
