Local Connectivity of the Filled Julia Set Bundle over the Mandelbrot Set

For the quadratic family $f_c(z)=z^2+c$, we study the non-escaping locus $\text{KM}=\{(c,z):c\in \text{M},z\in K_c\}$, the model space for cubic capture straightening in the framework of Inou and Kiwi. Here $\text{M}$ is the Mandelbrot set, $K_c$ is the filled Julia set of $f_c$, and $J_c=\mathrm{\partial }{K}_c$. We prove that $\mathrm{\partial }\text{KM}$ has full Hausdorff dimension $4$. The set $\text{KM}$ is locally connected at every $(c,z)$ with hyperbolic $c\in \text{M}$. At parameters on $\mathrm{\partial }\text{M}$, we construct mixed puzzle pieces from Yoccoz puzzles and parapuzzles. For non-degenerate pairs, shrinking of both planar puzzles implies local connectedness of $\text{KM}$ at $(c,z)$. However, local connectedness fails at some $(c,z)$ with $c$ real parabolic and $z\in K_c\setminus J_c$, even though both the parameter space and the filled Julia fiber are locally connected there.