Strongly self-dual graph realization conjecture for Reuleaux polyhedra

A planar, 3-connected graph $G$ is strongly self-dual if it has a self-duality isomorphism $\tau:G\to G^*$ such that, for every pair of vertices $u,v$, $u\in\tau(v)$ if and only if $v\in\tau(u)$, and no vertex satisfies $u\in\tau(u)$. A Reuleaux polyhedron is a ball-polyhedron in $\mathbb{E}^3$, that is, an intersection of finitely many unit balls, with each center and each vertex corresponding to the other. Strongly self-dual realization conjecture. For every planar, 3-connected, strongly self-dual graph $G$, there is a Reuleaux polyhedron in $\mathbb{E}^3$ whose edge graph is isomorphic to $G$. A weaker realization result provides a possibly non-injective point assignment; injectivity is equivalent to the conjecture, which the source identifies as open.