We investigate whether explicit models of warped D-brane inflation are possible in string compactifications. To this end, we study the potential for D3-brane motion in a warped conifold that includes holomorphically embedded D7-branes involved in moduli stabilization. The presence of the D7-branes significantly modifies the inflaton potential. We construct an example based on a very simple and symmetric embedding due to Kuperstein, z1= const, in which it is possible to fine-tune the potential so that slow-roll inflation can occur. The resulting model is rather delicate: inflation occurs in the vicinity of an inflection point, and the cosmological predictions are extremely sensitive to the precise shape of the potential.