Does pairwise independence and same distribution imply trivial Invariant $sigma$-algebra?

I know that by Kolmogorov’s $0-1$ Law, that for independent r.v, the tail $sigma$-algebra is trivial (e.g all events have probability either $0$ or $1$). This coupled with the ergodic theorem, one can easily derive the Strong Law of Large Numbers for $X_i$ i.i.d. and finite expected value.

I also know that there exists a stronger SLLN called Etemadi’s SLLN, which only requires finite expected value, and that $X_i$ have the same distribution and are pairwise independent.

With this in mind, I was wondering if pairwise independence and same distribution imply trivial Invariant $sigma$-algebra? If it does, can anyone provide a proof or a reference to such proof? And if no, can one provide a counter-example?