The simultaneous preperiodicity conjecture for 0 and 1 in the quadratic family

For $c\in\mathbb{C}$, let $f_c(z)=z^2+c$, and let $S_{0,1}$ denote the set of parameters $c$ for which both $0$ and $1$ are preperiodic under iteration of $f_c$. Simultaneous preperiodicity conjecture.

$$S_{0,1}=\{0,-1,-2\}.$$

The conjecture gives an exact description of the parameters for which the two marked points are simultaneously preperiodic. The stated computational evidence checks relations of bounded orbit length, but the source provides no resolution of the conjecture.