On a rigidity criterion for del Pezzo fibrations over ${\mathbb P}^1$

We discuss the rigidity problem for Mori fibrations on del Pezzo surfaces of degree 1, 2 and 3 over ${\mathbb P}^1$ and formulate the following conjecture: such a del Pezzo fibration $V/{\mathbb P}^1$ is birationally rigid if and only if its quasi-effective and adjunction thresholds coincide. We prove the "only if" part of this conjecture.