The tree cover bound for connected triangle-free graphs

Let $G$ be a connected triangle-free graph on $n$ vertices, and let $T(G)$ denote its tree cover number.

Triangle-free tree cover conjecture. For all connected triangle-free graphs,

$$T(G)\leq\left\lceil\frac{n}{3}\right\rceil.$$

The question arises because a related bound for graphs of girth at least five does not extend to the four-cycle; the conjecture is motivated by computations in Sage and remains open in the supplied source.