Elliptic fibrations on Fano threefold hypersurfaces in weighted projective space

The 95 families of quasismooth anticanonically embedded weighted Fano 3-fold hypersurfaces with terminal singularities were introduced by A.R. Iano-Fletcher and M. Reid in late eighties. A general hypersurface in each of the 95 families is proved to be birationally rigid by A. Corti, A. Pukhlikov, and M. Reid. This means that it can be birationally transformed into neither a conic bundle nor a del Pezzo fibration. In my talk, I will answer to the following question: Can a general hypersurface in each of the 95 families be birationally transformed into an elliptic fibration?