A new convergence problem for the conditional expectation

Question

You have risks $X_1$, $X_2$, ...

(they are assumed to be independent, but not necessarily identically distributed)

and

$S_n= X_1 + X_2 + \cdots +X_n$

QUESTION: under what reasonable conditions we have that

$\mathbb{E}[Xi | S_n]$ converges to $\mathbb{E}[Xi]$?