  Let $W_N(\beta) = \mathrm{E}_0\left[e^{ \sum_{n=1}^N \beta \omega(n,S_n) -
N\beta^2/2}\right]$ be the partition function of a two-dimensional directed
polymer in a random environment, where $\omega(i,x), i\in \mathbb{N}, x\in
\mathbb{Z}^2$ are i.i.d. standard normal and $\{S_n\}$ is the path of a random
walk. With $\beta=\beta_N=\widehat{\beta} \sqrt{\pi/\log N}$ and
$\widehat{\beta}\in (0,1)$ (the subcritical window), $\log W_N(\beta_N)$ is
known to converge in distribution to a Gaussian law of mean $-\lambda^2/2$ and
variance $\lambda^2$, with $\lambda^2=\log ((1-\widehat\beta^2)^{-1})$
(Caravenna, Sun, Zygouras, Ann. Appl. Probab. (2017)). We study in this paper
the moments $\mathbb E [W_N( \beta_N)^q]$ in the subcritical window, and prove
a lower bound that matches for $q=O(\sqrt{\log N})$ the upper bound derived by
us in Cosco, Zeitouni, arXiv:2112.03767 [math.PR]. The analysis is based on
appropriate decouplings and a Poisson convergence that uses the method of ''two
moments suffice''.
