Examples of Fano varieties of index one that are not birationally rigid

A conjecture of Pukhlikov states that a smooth Fano variety of dimension at least 4 and index one is birationally rigid. We show that a general member of the linear system given by the ample generator of the Picard group of the moduli space of stable, rank 2 bundles with fixed determinant of odd degree on a curve of genus at least 3 is not birationally rigid.