The Sibirsky ideal component conjecture

Consider the $1:-1$ resonant polynomial system

$$\dot x=x\left(1-\sum_{(i,j)\in S}a_{ij}x^iy^j\right),\qquad \dot y=-y\left(1-\sum_{(i,j)\in S}b_{ji}x^jy^i\right).$$

Let $I_S$ be its Sibirsky ideal and let $B$ be the Bautin ideal, with ${\bf V}(I_S)$ and ${\bf V}(B)$ their varieties. Sibirsky ideal component conjecture. The set ${\bf V}(I_S)$ is a component of the variety ${\bf V}(B)$; that is, it is a proper irreducible subvariety of ${\bf V}(B)$. This conjecture asserts that the Sibirsky ideal captures a genuine component of the integrability variety; the supplied text gives no resolution status.