Quadratic-field preperiodic graph conjecture

Let $K/\mathbb{Q}$ be a quadratic field and let $f\in K[z]$ be a quadratic polynomial. Let $\operatorname{PrePer}(f,K)$ be the set of $K$-rational preperiodic points of $f$, and let $G(f,K)$ be its directed preperiodic graph. Quadratic-field preperiodic graph conjecture. One has

$$\#\operatorname{PrePer}(f,K)\leq 15.$$

Moreover, the only graphs that may occur, up to isomorphism, as $G(f,K)$ are the $46$ graphs appearing in the paper's appendix.

The conjecture is based on extensive computations over quadratic fields and seeks a complete classification of preperiodic graphs for quadratic polynomials over such fields. It remains open.