Finiteness conjecture for irreducible polyhedral graphs with fixed rigid vertices

Let ${\mathcal{P}}_k^\text{irr}$ be the set of all irreducible polyhedral graphs with $k$ rigid vertices. Finiteness conjecture. Each set ${\mathcal{P}}_k^\text{irr}$ is finite. This conjecture asserts that, unlike reducible polyhedra, irreducible polyhedral graphs cannot have an unbounded number of non-rigid vertices while the number of rigid vertices remains fixed; the source indicates that this prediction is supported by further classifications but does not establish it.