Verstraëte's girth-six conjecture for independent domination in cubic graphs

Let $G$ be a cubic graph, and let $i(G)$ denote its independent domination number, the minimum size of an independent dominating set of $G$. The girth of $G$ is the length of a shortest cycle in $G$.

Verstraëte's conjecture. If $G$ is a cubic graph with girth at least $6$, then

$$i(G) \leq \frac{|V(G)|}{3}.$$

This conjecture seeks a sharper upper bound for independent domination in cubic graphs when short cycles, and in particular the complete bipartite extremal examples, are excluded. The source gives no resolution, so the conjecture remains open.