Nonemptiness and infinitude conjecture for geometric-progression value sets

Let $f(x,y)=(y^2-x^3)/x$. For $Q\in\mathbb{N}_{\geq 2}$, let $C_Q$ be the set defined in the paper for this rational function and quotient $Q$. Nonemptiness and infinitude conjecture. For each $Q\in\mathbb{N}_{\geq 2}$, the set $C_Q$ is nonempty and infinite. This predicts that every integer quotient $Q\geq2$ occurs and that each such quotient occurs for infinitely many parameters; the claim is supported in the source by numerical computations.