Integral $S_n$-invariant F-nef divisors are twice $G$-base-point-free

Let $\overline{\mathrm{M}}_{0,n}$ be the moduli space of genus $0$ stable pointed curves, and let $G$ denote the group used in the source to define $G$-base-point-freeness. Let $D$ be an integral $S_n$-invariant F-nef divisor on $\overline{\mathrm{M}}_{0,n}$.

Base-point-freeness conjecture. For any integral $S_n$-invariant F-nef divisor $D$, $2D$ is $G$-base-point-free; in particular, $2D$ is base-point-free.

In the source this statement appears as a computational theorem, with the hypothesis $n\leq 16$ stated immediately before it. Thus it is a solved bounded-range result rather than an open conjecture, and the claim above is understood with $n\leq 16$ over $\operatorname{Spec}\mathbb{Z}$.