Symmetric and non-symmetric F-conjectures are equivalent

The F-conjecture gives a conjectural description of the ample cone of the Deligne-Mumford moduli space $\overline{M}_{g,n}$. We prove that the $S_n$-symmetric and the non-symmetric F-conjectures are equivalent. We also prove the Strong F-conjecture for $\overline{M}_{0,8}$ (and give an alternative proof for $\overline{M}_{0,7})$. Finally, we derive, as a consequence, the F-conjecture for the moduli space of stable curves $\overline{M}_{g}$ up to genus $g\leq 44$.