The non-rigidity conjecture for Fano threefolds of codimension and index at least 2

Let $X$ be a Fano 3-fold with $Q$-factorial terminal singularities, embedded via the paper's specified embedding in a weighted projective space. Suppose that the codimension and Fano index of $X$ are both at least $2$.

Non-rigidity and non-solidity conjecture. Then $X$ is not birationally rigid. Moreover, if the codimension is at least $3$, then $X$ is non-solid.

The conjecture is motivated by the known non-solidity results for codimension $2$ Fano threefolds and by the birational constructions and classifications discussed in the paper. Its status is not resolved in the supplied source context.