The rank inequality for units of the Burnside ring

Let $G$ be a finite group. Let $\Omega(G)$ be the Burnside ring, let $\Omega^*(G)$ denote its unit group, and let $\Omega_2(G)$ be the subring of monomial representations induced from linear representations of subgroups whose values are only $\pm1$. This ring has a basis indexed by conjugacy classes of pairs $(H,\lambda)$, where $\lambda$ is a linear character of $H$ with $\lambda(h)=\pm1$ for every $h\in H$, equivalently by conjugacy classes of pairs $(H,K)$ with $K\leq H$ and $|H:K|\leq2$. Rank inequality conjecture. One has

$$\operatorname{rk} \Omega^*(G)-1\leq\operatorname{rk} \Omega_2(G)-\operatorname{rk} \Omega(G).$$

The inequality proposes a lower bound on the rank of the subring of monomial representations detected by sign-valued linear characters for the rank of the unit group of the Burnside ring. The surrounding discussion presents it as a conjecture motivating the computational study of these ranks; no resolution is given here.