  Abstract polytopes are a combinatorial generalization of convex and skeletal
polytopes. Counting how many flag orbits a polytope has under its automorphism
group is a way of measuring how symmetric it is. Polytopes with one flag orbit
are called regular and are very well known. Polytopes with two flag orbits
(called 2-orbit polytopes) are, however, way more elusive. There are $2^n-1$
possible classes of 2-orbit polytopes in rank (dimension) $n$, but for most of
those classes, determining whether or not they are empty is still an open
problem. In 2019, in their article An existence result on two-orbit maniplexes,
Pellicer, Poto\v{c}nik and Toledo constructed 2-orbit maniplexes (objects that
generalize abstract polytopes and maps) in all these classes, but the question
of whether or not they are also polytopes remained open. In this paper we use
the results of a previous paper by the author and Hubard to show that some of
these 2-orbit maniplexes are, in fact, polytopes. In particular we prove that
there are 2-orbit polytopes in all the classes where exactly two kinds of
reflections are forbidden. We use this to show that there are at least
$n^2-n+1$ classes of 2-orbit polytopes of rank $n$ that are not empty. We also
show that the maniplexes constructed with this method in the remaining classes
satisfy all but (possibly) one of the properties necessary to be polytopes,
therefore we get closer to proving that there are 2-orbit polytopes in all the
classes.
