On the cone of effective 2-cycles on $\overline{M}_{0,7}$

Fulton's question about effective k-cycles on (M) over bar (0),(n) for 1 < k < n - 4 can be answered negatively by appropriately lifting to (M) over bar (0),(n) the Keel-Vermeire divisors on (M) over bar (0), (k+1). In this paper we focus on the case of 2-cycles on (M) over bar (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 (M) over bar (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.