Nonrepresentability of acceptable rigidity-matroid constructions

Prove that if a graph G on n vertices yields an abstract 2-rigidity matroid M_G(R_{2,n}) and G is not isomorphic to K_{3,3}, then M_G(R_{2,n}) is not linearly realizable over any field of characteristic 0.

Background

The paper proves that the triangular-prism construction M_G(R_{2,6}) is not linearly representable over any characteristic-zero field. It proposes that this obstruction is general for all acceptable graphs except K_{3,3}, which is singled out as the apparent exceptional case. The conjecture seeks a complete characterization of linear realizability within this construction.