  Fulton's question about effective $k$-cycles on $\overline{M}_{0,n}$ for
$1<k<n-4$ can be answered negatively by appropriately lifting to
$\overline{M}_{0,n}$ the Keel-Vermeire divisors on $\overline{M}_{0,k+1}$. In
this paper we focus on the case of $2$-cycles on $\overline{M}_{0,7}$, and we
prove that the $2$-dimensional boundary strata together with the lifts of the
Keel-Vermeire divisors are not enough to generate the cone of effective
$2$-cycles. We do this by providing examples of effective $2$-cycles on
$\overline{M}_{0,7}$ that cannot be written as an effective combination of the
aforementioned $2$-cycles. These examples are inspired by a blow up
construction of Castravet and Tevelev.
