Nonexistence conjecture for quadratic 5-periodic points

Let $\mathbb{Q}_{\mathrm{quad}}$ denote the union of all quadratic number fields, equivalently the set of algebraic numbers satisfying an irreducible quadratic equation over $\mathbb{Q}$. For $c\in\mathbb{Q}$, consider the quadratic polynomial $\phi_c(z)=z^2+c$.

Nonexistence conjecture. There are no rational values $c$ such that $\phi_c$ has a 5-periodic point in $\mathbb{Q}_{\mathrm{quad}}$.

The claim is presented as a consequence of the Galois conjecture together with the known nonexistence of rational 5-cycles, and is also supported by the paper's independent computational and geometric investigation. The source gives no resolution status.