Akbari et al.'s upper-bound conjecture for edge Roman domination

Let $G$ be a simple graph of maximum degree $\Delta$ on $n$ vertices. An edge Roman dominating function is a function $f\colon E(G)\to\{0,1,2\}$ such that every edge assigned $0$ is adjacent to an edge assigned $2$, and the edge Roman domination number $\gamma'_R(G)$ is the minimum weight $\sum_{e\in E(G)}f(e)$ of such a function.

Akbari et al.'s conjecture.

$$\gamma'_R(G)\leq\left\lceil\frac{\Delta}{\Delta+1}n\right\rceil.$$

This conjecture seeks an upper bound for edge Roman domination in terms of the order and maximum degree of the graph, rather than its number of edges. The supplied text gives the preceding weaker bound $\gamma'_R(G)\leq\frac{2\Delta}{2\Delta+1}n$ but provides no resolution of the conjecture.