Hamiltonian torus actions and the unimodality of odd Betti numbers

This paper is dedicated to the question: Is the sequence of odd and even Betti numbers of a closed symplectic manifold with a non-trivial Hamiltonian torus action unimodal? Recently, there was some progress on the question for the sequence of even Betti numbers by Cho-Kim and the author. The results of this paper give positive evidence in the case of odd Betti numbers, in dimensions $6,8$ and $10$ under progressively stronger symmetry assumptions.