Milnor's homeomorphism conjecture for the parabolic Mandelbrot set

Let $\mathbf M$ be the Mandelbrot set of quadratic polynomials, and let $\mathbf M_1$ be the connectedness locus in the family of rational maps with multiplier $1$ at a fixed point. Milnor's conjecture. The parabolic Mandelbrot set $\mathbf M_1$ is homeomorphic to the Mandelbrot set $\mathbf M$. Moreover, the homeomorphism preserves the dynamics on the Julia sets. This conjecture asks whether the parameter space and Julia-set dynamics of the parabolic family can still be related to those of the classical Mandelbrot set, despite the failure of the usual quasiconformal deformation and straightening theories at the parabolic parameter. The source gives no resolution.