UMP test for a beta-type density

Let $X_1, X_2, \dots, X_n$ ($n > 1$) be a random sample from the following probability density function $f_\beta(x) = \begin{cases} \beta e^{-x} (1 - e^{-x})^{\beta - 1}, & x > 0 \ 0, & \text{otherwise} \end{cases}$, where $\beta > 0$ is an unknown parameter. For testing the following hypotheses, $H_0: \beta = 1$ against $H_1: \beta > 1$, at level $\alpha \in (0, 1)$, which of the following statements is correct?