Maniplexes with automorphism group ${\text{PSL}}_2(q)$

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 ${\text{PSL}}_2(q)$ for infinitely many prime powers $q$, and that no reflexible maniplex of rank $n>4$ exists that has ${\text{PSL}}_2(q)$ as its full automorphism group.