Burger and Mozes constructed (Burger and Mozes - Lattices in products of trees) infinite, finitely presented, torsion-free simple groups which split as amalgams of two finitely generated free groups over a finite index subgroup embedded by specific embeddings obtained through studying actions on product of regular trees. It is an interesting problem to make this amalgam as small as possible. For example, $F_{217} ∗_{F_{75601}} F_{217}$ is the smallest example constructed by Burger and Mozes; this was later improved to $F_9 \ast_{F_{81}} F_9$ by Rattaggi, and very recently to $F_5 \ast_{F_{25}} F_5$ by Caprace-Radu. These examples are crucially based on non-abelian finite simple groups, and in particular, the fact that $A_6$ is a non-abelian finite simple group such that the stabilizer of one letter in the action on $\{1, \dots, 6\}$ is isomorphic to another non-abelian finite simple group, namely $A_5$. For more details on this, see Rattaggi's excellent compendium.

My question, then, concerns the following: is this the smallest (with respect to some ordering on the ranks of the factors and the amalgamated subgroup) possible example of such a Burger-Mozes amalgam? It is clearly (?) not doable using only the original techniques of Burger and Mozes, as $A_5$ is of course the smallest non-abelian finite simple group.

I am also interested in any extensions of the original arguments to more general settings, e.g. by extending their normal subgroup theorem to a broader class of lattices.

Note that the ordering chosen on the factors and the amalgamated subgroup may have some relevance. Furthermore, some amalgams are isomorphic; the original Burger-Mozes example, for example, satisfies $F_{217} ∗_{F_{75601}} F_{217} \cong F_{349} ∗_{F_{75865}} F_{349}$.