Connected domination conjecture for plane triangulations

Let $G$ be a plane triangulation of sufficiently large order $n$. A connected dominating set is a subset $X\subseteq V(G)$ such that $V(G)=\bigcup_{x\in X}N_G[x]$ and $G[X]$ is connected; let $\gamma_c(G)$ denote the minimum cardinality of a connected dominating set.

Connected domination conjecture.

$$\gamma_c(G)\le\frac{3n}{8}.$$

The paper constructs plane triangulations with $\gamma_c(G)\ge 3|V(G)|/8-1$, so this weaker bound is presented as a revised conjecture; the supplied status evidence says it remains open.