Eventual characteristic alternating quotients of the rank-two free group

Let $F_2$ be the free group on two generators, and let $A_n$ denote the alternating group on $n$ elements. A finite quotient of $F_2$ is characteristic when it is of the form $F_2/C$ for a characteristic subgroup $C\leq F_2$. Eventual characteristic alternating quotient conjecture. There is an integer $N$ such that for all $n\geq N$, the alternating group $A_n$ is a characteristic quotient of $F_2$. The theorem immediately preceding this conjecture proves that infinitely many alternating groups occur as characteristic quotients of $F_2$, while the conjecture asks for all sufficiently large degrees.