Conjecture on generators of the simple loopless regular matroid complex homology

Let $(\mathcal{M}_{\bullet}^{\operatorname{\textbf{reg}},\operatorname{\textbf{simp}}},\partial_{\operatorname{del}})$ be the simple loopless regular matroid complex, and let its indecomposable quotient be the connected quotient complex $(\mathcal{M}_{\bullet}^{\operatorname{\textbf{reg}},\operatorname{\textbf{simp}},\operatorname{cont}},\partial_{\operatorname{del}})$. Write $U_{1,1}$ for the rank-one matroid on one element, let $W_{2k+1}$ be the wheel graph of genus $2k+1$ with $2(2k+1)$ edges, and let $\mathsf{M}(W_{2k+1})$ be its graphic matroid.

Generator conjecture. The homology of the indecomposable quotient of the simple loopless regular matroid complex is generated by the classes of $U_{1,1}$ and $\mathsf{M}(W_{2k+1})$ for all $k\geq1$.

Computations through $n\leq14$ find one-dimensional homology in degrees $(1,1)$, $(6,3)$, $(10,5)$, and $(14,7)$, generated respectively by $U_{1,1}$, $\mathsf{M}(W_3)$, $\mathsf{M}(W_5)$, and $\mathsf{M}(W_7)$. The conjecture predicts that this pattern continues for every odd wheel genus.