Let $M$ a reductive monoid, i.e. a integral normal affine scheme, which is a monoid whose group of units is a connected reductive group.

By Rittatore http://www.cmat.edu.uy/cmat/docentes/alvaro/publicaciones/pdfs/cohen.pdf, we know that they're all Cohen-Macaulay.

Do we know which of them are Gorenstein?

For example, is Vinberg's monoid (aka envelopping semigroup) Gorenstein, or at least the toric variety associated to it?