A linear bound for zero-sum Ramsey numbers over composite moduli

Let $k$ be a positive integer, let $c_k=c(k)$ be an integer, and let $G$ be a graph on $n$ vertices such that $k\mid E(G)$. The composite-modulus linear-bound conjecture. There is an integer $c_k=c(k)$ such that, for all $n$ and all graphs $G$ on $n$ vertices,

$$R(G,\mathbb{Z}_k)\leq n+c_k.$$

The preceding conjecture concerns prime moduli, whereas this proposal asks for the analogous uniform linear bound for every positive modulus. The constant is allowed to depend on $k$ but not on $n$ or on the graph $G$.