  We show that $sl_2$ 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.
