Quadratic zero-sum block conjecture for two-valued sequences

Let $r$, $s$ and $k$ be positive integers such that $(r+s)$ divides $k$. A function $f:[n]\to\{-r,s\}$ has total sum $f([n])=0$ when the sum of its values over $[n]$ is zero. A ZS $k$-block is a set of $k$ consecutive integers whose $f$-sum is zero. Quadratic zero-sum block conjecture. There exists a constant $c(r,s)$ such that, if

$$n\geq \frac{rs}{(r+s)^2}k^2+c(r,s)k,$$

then every function $f:[n]\to\{-r,s\}$ with $f([n])=0$ contains a ZS $k$-block. This conjecture would extend the paper's zero-sum block theorem from the values $\{-1,1\}$ to arbitrary positive integers $r$ and $s$; the authors indicate that it should be provable by similar methods, but leave it open.