Total domination conjecture for plane triangulations

Let $G$ be a plane triangulation of order $n\ge 6$. A total dominating set is a subset $X\subseteq V(G)$ such that $V(G)=\bigcup_{x\in X}N_G(x)$, and $\gamma_t(G)$ denotes the minimum cardinality of a total dominating set.

Total domination conjecture.

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

This is a domination analogue of the induced-subgraph conjectures discussed in the paper; the supplied text does not state a resolution.