Weak baby multicorns dynamically distinct conjecture

Let $d$ be the degree of a multicorn, let $\omega$ denote the rotational symmetry used in the parameter space, and let $H_1$ and $H_2$ be hyperbolic components of odd periods $k_1$ and $k_2$ of $\mathcal{M}_d^*$. Let $h_1>0>h_2$ and $h_3>0>h_4$ be the critical Ecalle heights specified by the period-doubling bifurcations from $H_1$ and $H_2$, respectively. Weak baby multicorns dynamically distinct conjecture. If

$$\omega^i H_1\neq H_2\quad\text{for }i=1,2,\ldots,d+1,$$

then

$$(h_1,h_2)\neq(h_3,h_4).$$

Critical Ecalle heights are preserved by hybrid equivalences, so this would give a dynamical distinction between the corresponding baby multicorn-like sets; the supplied text does not state whether the conjecture is resolved.