The average-of-roots-of-unity integrality conjecture for functions on cyclic groups

Let $n$ be a positive integer, let $f:\mathbb{Z}_n\longrightarrow\mathbb{Z}_n$ be a function, and let $\mu_f^{a,b}$ denote the average associated with $f$ for $a,b\in\mathbb{Z}_n$. Average-of-roots-of-unity integrality conjecture. If $\mu_f^{a,b}$ is an algebraic integer for every $a,b\in\mathbb{Z}_n$, then there exist $\alpha,\beta\in\mathbb{Z}_n$ such that

$$f(x)\equiv \alpha x+\beta$$

for all $x\in\mathbb{Z}_n$. Equivalently, $f$ should be representable by a linear polynomial in $\mathbb{Z}_n[X]$. The conjecture extends the result proved in the paper for prime $n$, where integrality of all the relevant averages implies that $f$ is linear; its validity for general $n$ is left open, and the paper notes that nonlinear functions might conceivably satisfy the integrality condition.