The realizability conjecture for cutsets in polytopes

Let $P$ be a $d$-polytope with graph $G_P$, and let a cutset be a set of vertices selected for removal by a hyperplane. Properties 1 and 2 of Proposition are connectedness of the cutset and connectedness of its complement in $G_P$. Realizability conjecture. For $d\geq 3$, every cutset satisfying properties 1 and 2 of Proposition is realizable. The claim extends the preceding proved three-dimensional result to all dimensions $d\geq 3$; the source gives no resolution.