UMVUE of an exponential tail

Let $X_{1},X_{2}$ be a random sample of size $2$ from an $\operatorname{Exp}(\theta)$ distribution, where $\theta>0$ is an unknown parameter. Let $$T_{1}=\begin{cases}1,&X_{1}>2,\\0,&\text{otherwise,}\end{cases}\qquad\text{and}\qquad T_{2}=\max\Bigl\{0, \dfrac{X_{1}+X_{2}-2}{X_{1}+X_{2}}\Bigr\}.$$ Which of the following statements is/are true?