Self-dual realizability conjecture for four-dimensional strongly involutive cones

A polyhedral cone is a convex cone defined by finitely many linear inequalities, and a realization of such a cone is a concrete geometric representation with the prescribed face structure. A cone is combinatorially self-dual if its face lattice is isomorphic to its dual face lattice, and it is strongly involutive when this self-duality is induced by an involutive correspondence between faces.

Self-dual realizability conjecture. All $4$-dimensional strongly involutive combinatorially self-dual polyhedral cones have self-dual realizations.

The claim is suggested by semidefinite computations producing slack matrices with the required support and rank for all examined instances with up to $17$ vertices, although those numerical certificates do not constitute a rigorous proof. The general assertion remains open.