Caro–Hegarty–Morrison quadratic-order conjecture for arithmetic-progression zero sums

Let $r$ and $s$ be fixed positive integers. For a function $f:[n]\to\{-r,s\}$ with $f([n])=0$, let $M(r,s,k)$ denote the minimum threshold such that every sufficiently large $n$ forces a zero-sum $k$-term arithmetic progression. Caro–Hegarty–Morrison's conjecture. There are positive constants $c_1=c_1(r,s)$ and $c_2=c_2(r,s)$ such that

$$c_1k^2\le M(r,s,k)\le c_2k^2.$$

Thus the threshold for zero-sum arithmetic progressions has quadratic order in $k$. Existing results give superlinear lower bounds and quadratic upper bounds, but the matching quadratic lower bound remains open.