Modified Grinenko conjecture on birational rigidity of del Pezzo fibrations

Let $d$ be an integer with $1\leq d\leq 3$. Let $X\to\mathbb P^1$ be a $\mathfrak D$-semistable del Pezzo fibration of degree $d$, and assume that it satisfies the $K$-condition, meaning that $-K_X$ does not lie in the interior of the cone of mobile divisors. Modified Grinenko conjecture. The fibration $X\to\mathbb P^1$ is birationally rigid. This is proposed as a modified version of Grinenko's conjecture in the setting of $\mathfrak D$-semistable del Pezzo fibrations; the source gives no resolution of the claim.