Polyhedrality and extreme divisors of the blow-up of the Losev–Manin space

Let $\phi: \overline{M}_{0,7}\rightarrow \operatorname{Bl}_e \overline{LM}_7$ be the natural reduction map. Let the divisors in Tables 1, 2, and 3 denote the specified $S_7$-equivalence classes of divisors on $\overline{M}_{0,7}$. Polyhedrality conjecture. Over characteristic $0$ and all but finitely many prime characteristics, $\operatorname{Eff}(\operatorname{Bl}_e \overline{LM}_7)$ is polyhedral with $138$ extreme rays, generated by the $\phi$-pushforwards of divisors $S_7$-equivalent to those in the specified tables. This conjecture gives a finite description of the effective cone of the blow-up of the Losev–Manin space; the surrounding discussion reports a conjectural description with millions of facets, while the stated characteristic-dependent assertion remains unresolved in the supplied text.