Relations among uniform MOM estimators

Let $X_1, X_2, \dots, X_n$ be a random sample from a $U\left(\theta+\frac{\sigma}{\sqrt{3}}, \theta+\sqrt{3}\sigma\right)$ distribution, where $\theta\in\mathbb{R}$ and $\sigma > 0$ are unknown parameters. Let $\bar{X}=\frac{1}{n}\sum_{i=1}^n X_i$ and $S = \sqrt{\frac{1}{n}\sum_{i=1}^n (X_i - \bar{X})^2}$. Let $\hat{\theta}$ and $\hat{\sigma}$ be the method of moment estimators of $\theta$ and $\sigma$, respectively. Then, which one of the following statements is FALSE?