The extremal-ray conjecture for cyclic divisors on $\overline{M}_{0,n}$

Let $m\geq 5$ be prime, let $m\mid n$, and let $d_1=\cdots=d_n$. For the associated $\mathbb{Z}_m$-tuple, let $\mathfrak{D}(\mathbb{Z}_m,E_m;\vec d)$ be the divisor defined in equation (E:new-nef), and let the symmetric nef cone mean the nef cone of $\mathfrak{S}_n$-invariant divisor classes on $\overline{M}_{0,n}$. Extremal-ray conjecture. The divisor $\mathfrak{D}(\mathbb{Z}_m,E_m;\vec d)$ generates an extremal ray of the symmetric nef cone of $\overline{M}_{0,n}$. The analogous assertion is proved in the case $m=3$ and equal $d_i$, while the prime cases $m\geq 5$ are proposed as an extension; the source does not indicate a resolution.