Period classification conjecture for quadratic Hénon maps over the rationals

Consider the family of polynomial automorphisms of $\mathbb{A}^2$ over $\mathbb{Q}$ given by

$$f(x,y)=(y,y^2+c+x),$$

where $c\in\mathbb{Q}$, and let $N$ be the period of a rational periodic point.

Period classification conjecture. Over $\mathbb{Q}$, the map has no point of period $N$ except possibly when

$$N\in\{1,2,3,4,6,8\}.$$

The source identifies this as one of two readily falsifiable conjectures and notes that infinite families of examples would make the asserted list sharp. Its resolution is left open.