The conjecture is the following. If you can find a proof of it, please contact me!
Let $X_1, X_2, \ldots, X_n$ be i.i.d. continuous random variables with $E[X_i]=0,$ Var$[X_i]=1$ and $E[\log(|X_1|)] < \infty$. Then
$$n^{-1}\sum_{i=1}^n\log|X_i-\bar{X}_n|1_{\{X_i\ne\bar{X}_n;i=1,\ldots,n\}}\stackrel{L}{\longrightarrow}E[\log|X_1|].$$