Determinantal factorization conjecture for R-disjoint graphs

Let $G$ be an $R$-disjoint graph. Write $B(G)$ for its bipartite part, and for each odd cycle $C$ of $G$, let $R(C)$ denote the subgraph associated with $C$. Determinantal factorization conjecture. Then

$$\det A(G)=\det A(G[B(G)])\prod_{C}\det A(G[R(C)]).$$

The product is taken over all odd cycles $C$ of $G$. The paper proves this factorization for the broader class of BAB-graphs, so the stated conjecture is verified for the particular case of $R$-disjoint graphs.