Criteria for baby Julia sets in generalized McMullen parameter slices

Let $W$ be a region in a one-dimensional subfamily of the generalized McMullen parameter space, parameterized by $a$. Suppose one critical value $v_+(a)$ is topologically stable, with $W$ entirely contained in a hyperbolic component of its bifurcation locus. Assume that the orbit of $v_+(a)$ is bounded and that $v_+(a)$ lies in a baby polynomial Julia set inside its larger Julia set, with a region $U'_a$ containing that baby Julia set such that $f_a:U'_a\to U_a$ is polynomial-like. Suppose another critical value $v_-(a)$ eventually maps onto $v_+(a)$ for some parameter in $W$, so that $v_-(a)$ lies in a preimage copy of the baby Julia set containing $v_+(a)$. Criteria for baby Julia sets in parameter space. If, as $a$ loops around $\partial W$, $v_-(a)$ loops around the corresponding eventual preimage of $U'_a\setminus U_a$, then the bifurcation locus of $v_-$ inside $W$ sweeps out a homeomorphic copy of the baby polynomial Julia set associated with $v_+(a)$. This is proposed as an alteration of the Douady–Hubbard criteria for baby Mandelbrot sets; establishing the statement analytically remains open.