Generic period-two conjecture for quadratic Hénon maps over the rationals

For $\delta,c\in\mathbb{Q}$, consider the polynomial automorphism

$$f(x,y)=(y,y^2+c-\delta x).$$

A rational periodic point means a point in $\mathbb{Q}^2$ that is periodic under $f$.

Generic period-two conjecture. For all but finitely many $\delta\in\mathbb{Q}$, the $\mathbb{Q}$-rational periodic points of every such map, with $c\in\mathbb{Q}$, have period dividing $2$.

The source presents this as a readily falsifiable conjecture and notes that infinite families of examples would make it sharp. The claim remains open.