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 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 M_{0,8} (and give an alternative proof for M_{0,7}). Finally, we derive, as a consequence, the F-conjecture for the moduli space of stable curves M_{g} up to genus gleq 44.