Weak Fulton conjecture on the equivariant nef cone of the moduli space of stable pointed curves

Let $\overline{\mathbf{M}}_{0,n}$ be the moduli space of stable $n$-pointed genus-zero curves, let $A_{\alpha}$ be the divisor considered in the paper, and let $p_i$ denote the stated generators of the relevant $\mathfrak{s}_n$-equivariant nef cone. Weak Fulton conjecture. For $\alpha\leq\frac{2}{[n/3]+1}$, the face of the $\mathfrak{s}_n$-equivariant nef cone containing $A_{\alpha}$ is the simplex generated by $p_{[n/3]},\dots,p_{\lfloor n/2\rfloor}$. The claim is a weaker, equivariant version of the description of the nef cone suggested by Fulton's conjecture. The supplied text proves related results about the F-simplex, but does not establish the stated claim or give evidence of its resolution.