Birational rigidity conjectures for codimension-two Fano threefold WCIs

Let $X$ be a quasismooth Fano threefold weighted complete intersection of codimension $2$ and index $1$, belonging to a deformation family indexed by $I$. Let $I_{br}$, $I_F$, and $I_{dP}$ denote the families described in the preceding classification, and in the case of $X$ belonging to $I_F$, let $X'$ be the Fano threefold in part (2) of the cited theorem. Birational rigidity conjecture. (1) Any smooth complete intersection of a quadric and a cubic in $\mathbb P^5$ is birationally rigid. (2) Any quasismooth member $X$ belonging to a family indexed by $I_F$ is not birational to a Mori fibre space other than $X$ and $X'$. The first assertion concerns the remaining smooth quadric–cubic case, while the second predicts the complete list of Mori fibre spaces birational to members of the families indexed by $I_F$. The source presents these assertions as conjectural; no resolution is supplied here.