We study models of hybrid inflation in the framework of supergravity with superconformal matter .
F-term hybrid inflation is not viable since the inflaton acquires a large tachyonic mass .
On the contrary , D-term hybrid inflation can successfully account for the amplitude of the primordial power spectrum .
Is is a two-field inflation model which , depending on parameters , yields values of the scalar spectral index down to n _ { s } \simeq 0.96 .
Generically , there is a tension between a small spectral index and the cosmic string bound albeit , within 2 \sigma uncertainty , the current observational bounds can be simultaneously fulfilled .