$s{l}_2$ conformal block divisors and the nef cone of ${\bar{M}}_{0,n}$

We show that $s{l}_2$ conformal block divisors do not cover the nef cone of ${\bar{M}}_{0,6}$, or the $S_9$-invariant nef cone of ${\bar{M}}_{0,9}$. A key point is to relate the nonvanishing of intersection numbers between these divisors and F-curves to the nonemptiness of some explicitly defined polytopes. Several experimental results and some open problems are also included.