Show there is no UMP test for $N(\mu,\sigma^2)$

Question

Here's a testing problem.

$X_1, \cdots, X_n, indep, \sim N(\mu, \sigma^2)$ where $\mu, \sigma^2$ are unknown.

$H_0: \mu \le \mu_0$ vs $H_1: \mu > \mu_0$ where $\mu_0$ is specified.

I need to prove that there's no UMP level $\alpha \in (0, 1/2)$ test.

Any hints or comments you can give me?

Thanks in advance.