The proper rainbow triangle-packing conjecture for edge-colored graphs

Let $G$ be an edge-colored graph on $n$ vertices, let $e(G)$ denote its number of edges, and let $c(G)$ denote its number of colors. A proper $mK_3$ is a collection of $m$ vertex-disjoint rainbow triangles. The proper rainbow triangle-packing conjecture. For every integer $m\geq 1$, if $n\geq 5m+2$ and

$$e(G)+c(G)>\binom{n}{2}+mn-\binom{m+1}{2},$$

then $G$ admits a proper $mK_3$. This conjecture is motivated by constructions showing that the previously proposed bound with $n\geq 5m$ is false; the paper proves related sufficient conditions but leaves this sharper bound open.