Generic single-flex conjecture for twinned polyhedra

Let $P$ be a triangulated polyhedron with distinguished vertices $A,B,A',B'$ satisfying

$$AB=A'B',\qquad AB'=BA'.$$

Form the twinned polyhedron $T$ of $P$ around the equator $ABA'B'$, as in the construction described in the paper. Generic single-flex conjecture. For almost all such triangulated polyhedra $P$, the twinned polyhedron $T$ has a single flex. The preceding theorem proves the analogous conclusion when the edges of $P$ form an isostatic framework; the conjecture asserts that the imposed equalities do not affect the generic case.