The spanning-tree bound conjecture for weighted triangle-free graphs

Let $G$ be a weighted triangle-free graph, and let $T$ be a spanning tree of $G$. Spanning-tree bound conjecture. One should have

$$\operatorname{mac}(G)\geq \frac{w(G)}{2}+\frac{3w(T)}{8}.$$

The conjecture would determine the optimal value of the spanning-tree parameter for triangle-free graphs, improving the established lower bound $1/4\leq\theta\leq3/8$ to $\theta=3/8$.