Let $X_{ij}$ with $1\leq i<j\leq n$ (that are $X_{12},\dots, X_{1n},\dots,X_{(n-1)n}$) be ${n \choose 2}$ identically normal distributed $N(0,\sigma^2)$ such that $ \text{corr}(X_{ij},X_{rs})=\rho $ if $|\{i,j\}\cap\{r,s\}|=1$ and $0$ if $|\{i,j\}\cap\{r,s\}|=0$.

I'd like to estimate $\sigma^2$ if in the large sample setting. I have proved that $\hat\sigma^2=\frac1{{n\choose 2}}\sum_{i=1}^n\sum_{j=i+1}^nX_{ij}^2$ converges to $\sigma^2$ in probability. But, how can I know that this estimator is the best estimator that we can do? For example, in the efficiency setting, do we have $\sqrt{{n\choose 2}}(\hat\sigma^2-\sigma^2)\to N(0,\tau^2)$ for some $\tau$? Any help will be appreciated.