It's not a conjecture. Gromov himself writes in this paper (see Section 0.4) an example of an odd dimensional manifold with non-vanishing minimal volume, by constructing a lower bound involving the so-called simplicial volume (and a corresponding metric $g$).

This response is unfortunately not about simply-connected manifolds, I overlooked that assumption by the author of the question. So, this is not an answer. See my comment below.

Gromov writes in this paper (see Section 0.4) an example of an odd dimensional (not simply-connected) manifold with non-vanishing minimal volume, by constructing a lower bound involving the so-called simplicial volume (and a corresponding metric $g$).