Linear domination conjecture for graph covers

Let $F$ be a graph, and let $G$ be a $k$-fold cover of $F$, meaning that each vertex of $F$ has $k$ vertices lying above it in the cover. Write $4\gamma(F)$ and $4\gamma(G)$ for the domination numbers of $F$ and $G$, respectively.

Linear domination conjecture. There exists a constant $c>0$ such that for every $k$-fold cover $G$ of a graph $F$,

$$\gamma(G) \geqslant c k\gamma(F).$$

The conjecture proposes a uniform linear lower bound for the domination number of a graph cover in terms of its fold and the domination number of the base graph. The supplied text gives motivating examples but no resolution, so the conjecture remains open.