The colorful-complex extremal characterization conjecture

Let $G$ be a graph, let $\mathcal{U} = \{U_1, \ldots, U_m\}$ be a partition of $V(G)$, and let $k \ge -1$. Write $\mathbf{Col}(\mathcal{I}(G),\mathcal{U})$ for the colorful complex of independent transversals, and let $\widetilde{H}_j$ denote reduced homology over a fixed ring. Suppose that

$$\widetilde{H}_k(\mathbf{Col}(\mathcal{I}(G),\mathcal{U})) \neq 0,$$

while

$$\widetilde{H}_j(\mathbf{Col}(\mathcal{I}(G-U_i),\mathcal{U}-\{U_i\})) = 0$$

for every $U_i \in \mathcal{U}$ and every $-1 \le j \le k$, that $G$ is a disjoint union of complete bipartite graphs, and that $G$ has exactly $|\mathcal{U}|+k$ connected components. The colorful-complex extremal characterization conjecture. Then for every $U_i \in \mathcal{U}$, there exists a complete bipartite component $K_{A,B}$ of $G$ such that $A \subseteq U_i$.