Consistency — estimating a ratio in a shifted exponential

Let $X_1,X_2,…,X_n$ $(n\ge2)$ be a random sample from an Exponential distribution with the probability density function

$$f(x\mid\mu,\sigma)=\frac1\sigma\exp\left(\frac{\mu-x}{\sigma}\right) \text{if }x>\mu, \qquad 0 \text{otherwise},$$

where parameters $\mu$ and $\sigma$ are unknown and positive. Let $\bar X_n$, $S_n^2$ and $X_{1:n}$ denote the sample mean, the sample variance and the sample smallest order statistic, respectively and let $\theta=\frac{\sigma}{\mu}$. Then which of the following statements are true?