  A maniplex of rank $n$ is a combinatorial object that generalises the notion
of a rank $n$ abstract polytope. A maniplex with the highest possible degree of
symmetry is called reflexible. In this paper we prove that there is a rank $4$
reflexible maniplex with automorphism group $\textrm{PSL}_2(q)$ for infinitely
many prime powers $q$, and that no reflexible maniplex of rank $n > 4$ exists
that has $\textrm{PSL}_2(q)$ as its full automorphism group.
