Uniform integrability of the maximum of a random walk with negative drift

Question

Given $S_k^{(n)} = X_1^{(n)} + ... + X_k^{(n)}$ for all $k,n\in\mathbb{N}$, where the $X_i^{(n)}$'s are iid with mean $-\gamma$ for some $\gamma > 0$ and unit variance. Let \begin{equation}M^{(n)} = \max_k S^{(n)}_k\end{equation} I have so far proven that $M^{(n)}$ converges in distribution to a random variable $M$ as $n\rightarrow\infty$. However, I would also like to show that $\mathbb{E}M^{(n)}\rightarrow \mathbb{E}M$ for which I need the uniform integrability of $M^{(n)}$. Does this hold true under these conditions?