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 $β$ to depend on the length of the polymer $2n$, with $β=n^{-α}$ for some $α<1/4$. 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 $1/5<α<1/4$.