Necessity and sufficiency of D-unstable cores for network instability

A $D$-unstable core is a $D$-unstable Child-Selection matrix $\mathbb{S}[\pmb{\kappa}]$ such that none of its principal submatrices are $D$-unstable. A network is said to admit instability when it possesses such an instability-inducing mechanism. D-unstable core conjecture. A network admits instability if and only if it possesses $D$-unstable cores. The context states that $D$-unstable cores are sufficient for a network to admit instability; the conjecture is that they are also necessary. Its resolution is not established in the supplied text.