The $S_n$-invariant F-conjecture for $\overline{\mathrm{M}}_{0,n}$

Let $\overline{\mathrm{M}}_{0,n}$ be the moduli space of genus $0$ stable pointed curves, with the natural action of $S_n$ permuting the marked points. An effective divisor $D=\sum b_I B_I$ is F-nef if it has nonnegative intersection with every F-curve.

The $S_n$-invariant F-conjecture. An $S_n$-invariant divisor on $\overline{\mathrm{M}}_{0,n}$ is nef if and only if it is F-nef.

This is the restriction of the F-conjecture to the $S_n$-invariant divisor space and is the main conjectural formulation studied in the paper. The paper reduces it to a feasibility problem in polyhedral geometry and establishes related semi-ampleness results computationally for bounded $n$, but the conjecture remains open in general.