X-replacement and double-V-replacement preserve 3D independence

Prove that in the 3-dimensional rigidity matroid: (a) if v has neighbors {v1,…,v5}, v1v2, v3v4 ∉ E(G), and G − v + v1v2 + v3v4 is R_3-independent, then G is R_3-independent; and (b) if v1v2, v2v3, v_i v_j, v_j v_k ∉ E(G) for some v_i, v_j, v_k ∈ N_G(v) with v_j ≠ v2, and both G − v + v1v2 + v2v3 and G − v + v_i v_j + v_j v_k are R_3-independent, then G is R_3-independent.

Background

These operations are natural 3D analogues of Henneberg-type moves. They are known to preserve independence in the C_2^1-cofactor matroid, and would imply Whiteley’s conjecture and the 3D rank-cover conjectures via induction if established for R_3.

A proof would yield powerful inductive constructions for 3D rigid graphs.