Rescaling conjecture for the MCheb polynomial

Let $P_{\rm MCheb}(z):=z^2-1.87450961730020085\ldots$ be the real quadratic polynomial constructed so that its first-return combinatorics imitate those of the Chebyshev polynomial $f_{-2}(z)=z^2-2$. Consider the nest pieces of $P_{\rm MCheb}$ and the functions induced by their first return maps. Rescaling conjecture for the MCheb polynomial. There are suitable rescalings of the nest pieces of $P_{\rm MCheb}$ such that the induced functions converge to $f_{-2}$ and the properly rescaled pieces converge to the interval $[-2,2]$ in the Hausdorff topology. The conjecture asks whether the combinatorial analogy with the Chebyshev polynomial persists metrically, despite the lack of interior in the filled Julia set $K_{-2}$; the methods developed for $Q$-recurrent polynomials do not establish this behavior.