Is it true that \[\alpha_1(G)+\tau_1(G) \leq \frac{n^2}{4}?\]

OPEN

Let $G$ be a graph on $n$ vertices, $\alpha_1(G)$ be the maximum number of edges that contain at most one edge from every triangle, and $\tau_1(G)$ be the minimum number of edges that contain at least one edge from every triangle.

Is it true that \[\alpha_1(G)+\tau_1(G) \leq \frac{n^2}{4}?\]

A problem of Erdős, Gallai, and Tuza [EGT96], who observe that this is probably quite difficult since there are different examples where equality hold: the complete graph, the complete bipartite graph, and the graph obtained from $K_{m,m}$ by adding one vertex joined to every other.