Győri–Keszegh's greedy-partition triangle conjecture for $K_4$-free graphs

Let $G$ be an $n$-vertex $K_4$-free graph with $e$ edges. A greedy partition $P$ of $G$ is a partition of $V(G)$ into disjoint cliques $T_i$ such that $|T_i|\geq |T_{i+1}|$ for each $i$, and for every $\ell\geq 1$, the union of cliques of size at most $\ell$ induces a $K_{\ell+1}$-free subgraph. Write $r:=r(P)$ for the number of cliques in $P$, let $t(G)$ be the number of triangles in $G$, and let $t_e(G)$ be the maximum number of edge-disjoint triangles in $G$. Define

$$g(G,P):=r(e-r(n-r))-t(G).$$

Győri–Keszegh's conjecture. For every such graph $G$ and every greedy partition $P$,

$$t(G)\geq r\bigl(e-r(n-r)\bigr),\qquad\text{equivalently }g(G,P)\leq 0,$$

and consequently

$$t_e(G)\geq e-r(n-r).$$

The conjecture strengthens the known triangle bound used by Győri and Keszegh to derive the extremal result that every $K_4$-free graph with $t_{n,2}+m$ edges contains at least $m$ edge-disjoint triangles. A greedy-partition lemma gives $t_e(G)\geq t(G)/r(P)$, so the conjectured inequality would imply the stated lower bound for edge-disjoint triangles; its status is not resolved in the supplied text.