Semistability-modified birational rigidity conjecture for cubic del Pezzo fibrations

Let $X$ be a $dP_3/\mathbb{P}^1$, meaning a del Pezzo fibration of degree $3$ over $\mathbb{P}^1$, and suppose that $X$ is semistable in the sense of Kollár. A del Pezzo fibration satisfies the $K$-condition when its anticanonical divisor does not lie in the interior of the mobile cone.

Semistability-modified conjecture. If $X$ satisfies the $K$-condition, then $X$ is birationally rigid.

This is proposed as an updated form of Grinenko's conjecture after counterexamples in the singular degree-$3$ case. The source does not provide evidence resolving this semistability-restricted statement.