It is well known that a non-abelian free group is residually a finite simple group. Katz and Magnus proved, in fact, that non-abelian free groups are residually alternating and residually $PSL_{2}$. S. J. Pride has some nice results along these lines as well. The best result that I know of is the theorem of Weigel that can be formulated as follows. If $\mathfrak{X}$ is a group-theoretic class containing an infinite set of pairwise non-isomorphic finite non-abelian simple groups, then every non-abelian free group is residually an $\mathfrak{X}$-group.

My question is this:

Is a non-abelian free group fully residually a finite non-abelian simple group?

It seems likely that the answer to such an obvious question is known, but I have not been able to find it in the literature.

I should probably add that I suspect we can probably replace "finite non-abelian simple" with "alternating", but I haven't yet given any thought to the other infinite series. I'd like to learn whether anything is known before spending more time on this.