String C-group representations of alternating groups

We prove that for any integer $n\geq 12$, and for every $r$ in the interval $[3, \ldots, \lfloor (n-1)/2\rfloor]$, the group $A_n$ has a string C-group representation of rank $r$ therefore showing that the only alternating group whose set of ranks is not an interval is $A_{11}$.