Grinenko's K-condition conjecture for degree-1 del Pezzo fibrations

Let $X\to\mathbb{P}^1$ be a del Pezzo fibration of degree $1$ with at most terminal singularities. It satisfies the $K$-condition when $-K_X$ is not in the interior of the movable cone $\operatorname{Mov}(X)$. Grinenko's conjecture. The fibration $X/\mathbb{P}^1$ is birationally rigid over the base if and only if it satisfies the $K$-condition. The only-if direction is proved, while the converse remains open in this generality; the paper studies sufficient conditions for birational or superrigidity in the singular case.