The Bautin-variety invariance conjecture for polynomial vector fields

Consider the polynomial system denoted by (gs), its Bautin ideal $B$, and the associated parameter-space vector field denoted by (Vvf_sh). Bautin-variety invariance conjecture. The components of the variety of the Bautin ideal of system (gs) are invariant sets of system (Vvf_sh) defined as intersections of invariant algebraic surfaces of system (gs), where those surfaces are defined by homogeneous polynomials. The conjecture extends the preceding proposition from the displayed example to the general system; the supplied text gives no resolution status.