The asymptotic weighted domination bound for connected subcubic planar graphs

Let $G$ be a connected subcubic planar graph, and for $i\in\{0,1,2,3\}$ let $n_i(G)$ denote the number of vertices of degree $i$ in $G$. Let $\gamma(G)$ denote the domination number of $G$. The conjectured optimal bound. There exists a constant $N$ such that every such graph with at least $N$ vertices satisfies

$$\gamma(G) \leq n_0(G) + \frac{13}{20} n_1(G) + \frac{1}{2} n_2(G) + \frac{7}{20} n_3(G).$$

The conjecture is motivated by a constructed family showing that the coefficient $\frac{1}{2}$ on $n_2(G)$ would make the paper's upper bound tight. The source gives no resolution.