MVB estimator achieves the Cramer–Rao bound

Let $x_1,x_2,\dots,x_n$ be a random sample from $N(\mu,\sigma^2)$, the parameter $\mu$ being known. The unknown $(\theta=\sigma^2)$ parameter is $\sigma^2$ with parametric space $\Theta=\{\theta\mid0<\theta<\infty\}$ and $S_0^2=\tfrac1n\sum_{i=1}^n(x_i-\mu)^2$. Which of the following statements is/are correct?

I. $\operatorname{Var}(S_0^2)$ is more than the CR lower bound.

II. $\operatorname{Var}(S_0^2)$ coincides with the CR lower bound.

III. $S_0^2$ is an MVB estimator of $\theta$.