Consider probability distributions $P$ and $Q_n$ for $n \in \mathbb{N}$. For each $n$ and $\alpha \in [0, 1]$, the level-$\alpha$ optimal test between $H_0: X \sim P \text{ vs } H_1: X \sim Q_n$ is given as a likelihood ratio test by the Neyman-Pearson lemma, and let $\gamma(n, \alpha)$ be its type II error.
Now fix $\alpha \in [0, 1]$ and let $\gamma(\alpha)$ be the type II error of the level-$\alpha$ optimal test between $H_0: X \sim P \text{ vs } H_1: X \sim Q$. It can be shown that if $Q_n$ converges to $Q$ in total variation, $\gamma(n, \alpha)$ converges to $\gamma(\alpha)$ as $n \to \infty$.
The question is, under some weaker notion of convergence (for example, $Q_n \to Q$ in distribution), can we show that $\limsup_n \gamma(n, \alpha) \leq \gamma(\alpha)$?