Generalized Maximum Modulus Principle for cyclic groups

Fix $N>2$ and $d\geq 1$. Let $\Omega_N$ be the group of $N$th roots of unity, and let $f:\Omega_N^n\to\mathbf{C}$ have degree at most $d$ and local degree at most $N-1$, so that

$$f(z)=\sum_{\alpha\in\{0,1,\dots,N-1\}^n:\lvert\alpha\rvert\leq d}a_{\alpha}z^{\alpha}.$$

Generalized Maximum Modulus Principle. There exists $C(d,N)>0$, independent of $n$, such that for every $n\geq 1$,

$$\max_{z\in\operatorname{conv}(\Omega_N)^n}\lvert f(z)\rvert\leq C(d,N)\max_{z\in\Omega_N^n}\lvert f(z)\rvert.$$

For $N=\infty$ and $N=2$ the analogous inequality is trivial with constant $1$, whereas the case $2<N<\infty$ is described as highly non-trivial and motivates the conjecture.