Forest deletion conjecture for maximum 2-density

For a graph $G$, let $m(G)=\max_{G'\subseteq G,\,v(G')\geq 1}e(G')/v(G')$ denote its maximum average density, and let $m_2(G)$ denote its maximum $2$-density. Forest deletion conjecture. There exists a forest $F\subseteq G$ such that

$$m_2(G\setminus F)\leq m(G).$$

This conjecture is proposed as an independent statement whose resolution would imply the Kohayakawa–Kreuter conjecture in all cases.