Pukhlikov's birational rigidity conjecture for Fano varieties of index one

Let $V$ be a smooth Fano variety of dimension $\dim V\geq4$ whose Picard group is generated by the canonical class $K_V$. Pukhlikov's conjecture. If $\dim V\geq4$, then $V$ is birationally rigid; if $\dim V\geq5$, then $V$ is superrigid. The conjecture predicts birational rigidity, and in dimensions at least five birational superrigidity, under the stated Picard-group hypothesis. The paper's title indicates that it studies counterexamples to this prediction, so its status should be checked against the results of the paper and subsequent literature.