Ramification bound and attainment for discrete polynomial maps

Let $f$ be a polynomial of degree $k$ on a discrete Riemann surface, and let the ramification number of $f$ at a cycle $\gamma$ be its winding number around the origin, when defined. Ramification conjecture. The ramification number of a polynomial of degree $k$ is at most $k$; moreover, for every polynomial of degree $k$, there exists a cycle around which its ramification number is $k$. The statement generalizes the preceding observations for the discrete powers, but the supplied text gives no resolution.