Edge-size conjecture for graceful zillion graphs

Let $\Gamma$ be a zillion graph with $\omega$ odd cycles and $\mu$ even cycles, and let $|E(\Gamma)|$ denote its number of edges. Edge-size conjecture. There exist a quadratic polynomial $f\in\mathbb{Z}[x]$ and a linear polynomial $g\in\mathbb{Z}[x]$ such that $\Gamma$ is graceful whenever

$$|E(\Gamma)|\geq f(\omega)+g(\mu).$$

Kotzig's lower bound, as adapted in the source, shows that a zillion graph with $\omega$ odd cycles must satisfy a quadratic lower bound in its number of edges for gracefulness. The proposed conjecture gives a quadratic contribution from odd cycles and a linear contribution from even cycles, and remains open.