On additive/multiplicative property of permanent

Searching the literature brings up a concept known as permanent groups. A permanent group is a group of nonsingular matrices on which the permanent is multiplicative. In was conjectured by M. Marcus in Permanents that the set nonsingular matrices of the form $PD$, where $P$ is a permutation matrix and $D$ is a diagonal matrix, is a maximal permanent group (see Conjecture 12). This conjecture is proven over the complex numbers by Beasley in Maximal groups on which the permanent is multiplicative. Further work has been done by Beasley and Cummings in the papers titled Permanent Groups, Permanent Groups II, and Permanent Semigroups, etc.