Upper semicontinuity conjecture for the mother hedgehog

Let $c\in\partial\boldsymbol{\Delta}$ be a non-parabolic parameter, let $f_c$ be the corresponding quadratic polynomial, and let $H_c$ denote its mother hedgehog: if the closed Siegel disk contains a critical point, $H_c$ is that closed Siegel disk; otherwise, $H_c$ is the closure of the union of all hedgehogs of $f_c$. Mother hedgehog conjecture. The mother hedgehog $H_c$ depends upper semicontinuously on $c$. This is the indifferent-dynamical analogue of the upper semicontinuous dependence of the filled-in Julia set on the polynomial. Its resolution would provide continuity control for mother hedgehogs across non-parabolic parameters.