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 $\KM=\{(c,z):c\in\M,\ z\in K_c\}$, the model space for cubic capture straightening in the framework of Inou and Kiwi. Here $\M$ is the Mandelbrot set, $K_c$ is the filled Julia set of $f_c$, and $J_c=\partial K_c$. We prove tha