Well Total Domination of Triangle-Free Graphs with Bounded Induced Path Size

Let $G$ be a finite simple connected graph with at least two vertices. Suppose that $G$ is triangle-free, meaning $G$ contains no three distinct vertices $a, b, c$ with edges $ab$, $bc$, and $ca$. Suppose further that the size of a largest induced path in $G$ is at most $4$. A subset $S \subseteq V(G)$ is an induced path if the induced subgraph $G[S]$ is a tree in which every vertex has degree at most $2$ (equivalently, $G[S]$ is a path graph); the size of a largest induced path is the maximum cardinality of such a subset. A graph is well totally dominated if every minimal total dominating set has the same cardinality. Prove that $G$ is well totally dominated.