Triangle-covering conjecture for graphs above the Mantel threshold

Let $G$ be an $n$-vertex graph, let $T(G)$ denote its number of triangles, and let $\tau_{\triangle}(G)$ be the minimum size of a vertex set meeting every triangle. For integers $0<t<s$ and sufficiently large $n$, consider the graph $K_{\lceil n/2\rceil,\lfloor n/2\rfloor}^{s,t}$ obtained from the complete bipartite graph by adding $s-1$ pairwise disjoint edges in the first part and one edge in the second part, then deleting $s-t$ edges incident with one endpoint of that second-part edge. Triangle-covering conjecture. If $G$ has $n$ vertices and $\left\lfloor\frac{n^2}{4}\right\rfloor+t$ edges, satisfies $\tau_{\triangle}(G)\geq s$, and $n\geq n(t,s)$ is large, then $G$ contains at least as many triangles as $K_{\lceil n/2\rceil,\lfloor n/2\rfloor}^{s,t}$, namely

$$(s-1)\left\lfloor\frac n2\right\rfloor+\left\lceil\frac n2\right\rceil-2(s-t).$$

The construction shows the proposed bound is attainable for sufficiently large $n$; the lower-bound assertion itself is left as a conjecture.