Transversal-matroid characterization of minimally angle-rigid colored graphs

Let $(G,c)$ be a colored graph, let $T_{c,j}(G)$ denote the family of transversal edge sets associated with color $j$, and let $H$ be a minimally rigid graph. Transversal characterization conjecture. The graph $(G,c)$ is minimally angle-rigid if and only if, for each color $j$, there exists a transversal edge set $X \in T_{c,j}(G)$ and a minimally rigid graph $H$ such that

$$G=H\cup X,\qquad X\cap E(H)=\emptyset.$$

This conjecture is the formal version of the paper's proposed transversal-matroid description, motivated by experimental data and heuristic arguments. No resolution is supplied in the source, so the conjecture remains open.