Consider the following Hypothesis Testing problem:
Hypothesis $H_0$ : $X \sim N(\mu_0, \sigma_0)$. Mean $\mu_0$ is known but only upper and lower bounds on $\sigma_0$ are known.
Hypothesis $H_1$ : $X \sim N(\mu_1, \sigma_1)$. Mean $\mu_1$ is known but only upper and lower bounds on $\sigma_1$ are known.
How could one solve this problem? Do uniformly most powerful tests exist for this problem?