Conder's stronger conjecture on alternating quotients of amalgamated free products

Let $A$ and $B$ be finite groups, let $C$ be a subgroup of $A\cap B$ of index at least $2$ in $A$ and at least $3$ in $B$, and let $K$ be the core of $C$ in the amalgamated free product $A\ast_C B$. The Conder conjecture. All but finitely many alternating groups occur as the image of $A\ast_C B$ under some homomorphism that takes $A$ and $B$ to subgroups of the alternating group isomorphic to $A/K$ and $B/K$, respectively. This is stated as a stronger conjecture of Marston Conder concerning quotients that preserve the specified subgroup types; the supplied source gives no resolution.