Bi-rigidity conjecture for standard degree-three del Pezzo fibrations

Let $X$ be a standard degree-three del Pezzo fibration, meaning a standard del Pezzo fibration whose generic fibre is a cubic surface. Bi-rigidity conjecture. The threefold $X$ is bi-rigid. The conjecture follows a known nonrationality result for a general member of the relevant linear system and concerns the birational behaviour of standard cubic-surface fibrations; its general validity remains open.