Quadratic mating conjecture

Let $P$ and $Q$ be quadratic polynomials with locally connected Julia sets, and let $p$ and $q$ denote their corresponding parameters in the Mandelbrot set. The quadratic mating conjecture. The geometric mating of $P$ and $Q$ exists unless $p$ and $q$ lie in conjugate limbs of the Mandelbrot set. The postcritically finite case is known by the Rees–Shishikura–Tan theorem for polynomials not from conjugate limbs; the conjecture concerns the general locally connected case, where uniqueness and existence are not obvious, particularly in the presence of invariant line fields.