The unique-centre conjecture for simplices

Let a simplex be a geometric simplex, and call a point a simplex centre if it is assigned to the simplex and is fixed by every isometry of the ambient space that leaves the simplex invariant. A simplex is equifacetal if all of its facets, its maximal proper faces, are congruent to one another.

Unique-centre conjecture. If a simplex has a unique simplex centre, then it is equifacetal.

The paper proves the converse implication: every equifacetal simplex has a unique simplex centre, using transitivity of its isometry group on the vertices. The proposed converse remains open in the supplied text.