Character-table characterization of totally extremal ideal Perron similarities

A normalized ideal Perron similarity is a similarity matrix with the normalization and ideal Perron properties defined in the surrounding paper. It is totally extremal when every entry associated with its Perron-similarity polytope is an extremal point.

Character-table conjecture. If $S$ is a normalized ideal Perron similarity and $S$ is totally extremal, then $S$ is the character table of a finite Abelian group.

This proposes that total extremality characterizes character tables of finite Abelian groups among normalized ideal Perron similarities. The source presents it as a question for further inquiry; no resolution is supplied here.