Rational-function lower-bound conjecture

Let $q$ be a positive integer, let $1\leq u_1<u_2<\cdots<u_n<q$ with $\gcd(u_k,q)=1$ for every $k$, and let $v_1,\ldots,v_n$ be arbitrary integers. Define

$$f(x)=\sum_{k=1}^n\frac{x^{v_k}}{1-x^{u_k}}.$$

Rational-function lower-bound conjecture. If

$$f(e^{2\pi i/q})=0,$$

then

$$\sum_{k=1}^n u_k\geq q.$$

The conjecture is intended to rule out certain vanishing Fourier sums arising from Beatty coverings and would simplify the analysis of Fraenkel's conjecture. It is presented as an unproved auxiliary conjecture.