The large-girth planar edge Roman domination conjecture

Let $G$ be a planar graph of girth at least $3k+2$ with $n$ vertices, and let $\gamma'_R(G)$ denote its edge Roman domination number: the minimum of $\sum_{e\in E(G)}f(e)$ over all functions $f\colon E(G)\to\{0,1,2\}$ such that every edge assigned $0$ is adjacent to an edge assigned $2$. Large-girth planar conjecture. If $G$ is a planar graph of girth at least $3k+2$ on $n$ vertices, then

$$\gamma'_R(G)\leq \frac{2k+2}{3k+2}n.$$

The conjecture is motivated by known bounds for planar graphs of specified girth, and its right-hand coefficient tends to $\frac{2}{3}$ as $k$ tends to infinity, reflecting the tree-like behavior of planar graphs with large girth. No resolution is supplied in the given text.