Minimal forbidden-minor conjecture for the sign pattern $s_*$

Let $\Sigma \in \operatorname{PR}_5$ be a real symmetric matrix whose principal-minor sign pattern $\operatorname{signs}_\Sigma$ agrees with $s_*$ on all subsets of sizes $1$ and $3$. Minimal forbidden-minor conjecture. Then

$$\det \Sigma < 0.$$

In particular, $s_*$ is not representable. The preceding reductions show that representability can be tested using a polynomial system with ten variables, ten sign constraints on order-three principal minors, and the positivity of the determinant, but neither a representation nor a proof of nonrepresentability is currently known.