On chiral polytopes having a group PSL(3,q)$\mathrm{PSL}(3,q)$ as automorphism group

For each prime power q > 5$q\geqslant 5$, we construct a rank four chiral polytope that has a group PSL(3,q)$\mathrm{PSL}(3,q)$ as automorphism group and Schlafli type [q-1,2(q-1)(3,q-1),q-1]$[q-1,\frac{2(q-1)}{(3,q-1)},q-1]$. We also construct rank five polytopes for some values of q$q$ and we show that there is no chiral polytope of rank at least six having a group PSL(3,q)$\mathrm{PSL}(3,q)$ or PSU(3,q)$\mathrm{PSU}(3,q)$ as automorphism group.