Goldberg–Milnor conjecture on perturbing parabolic polynomial cycles

Let $P$ be a polynomial having a parabolic cycle, and let $J(P)$ denote its Julia set. A small perturbation of $P$ should convert the immediate basin of the parabolic cycle into an attracting basin, while the perturbed polynomial restricted to its Julia set is topologically conjugate to $P$ restricted to $J(P)$. Goldberg–Milnor conjecture. For any polynomial $P$ having a parabolic cycle, the immediate basin of the parabolic cycle can be converted to be attracting by a small perturbation, and the perturbed polynomial on its Julia set is topologically conjugate to the original polynomial $P$ on $J(P)$. This problem was proposed by Goldberg and Milnor. The surrounding discussion indicates that the paper's methods naturally extend to it, but does not state a resolution.