An orientation obstruction to connectedness of minimal positive semidefinite factorization orbits

We give a positive integer $6\times6$ matrix of ordinary rank six and real
positive semidefinite rank three whose space of minimal positive semidefinite
factorizations, modulo real congruence, is disconnected. A continuous
orientation sign is invariant under every size-three congruence and takes both
values on explicit positive definite factorizations. This gives a negative
answer to the connectedness question of Fawzi–Gouveia–Parrilo–Robinson–Thomas.
An intrinsic three-dimensional support orientation extends the construction to
every factor size at least three; a compactness argument also yields strictly
positive rational examples in every such size.