Extension of birational rigidity to mildly singular del Pezzo fibrations

A del Pezzo fibration is birationally rigid if it has only one Mori fiber space structure in its birational class. Let $X$ be a del Pezzo fibration satisfying the hypotheses of Theorem 1.2: in the smooth case, $\operatorname{Pic}(X)=\mathds{Z}\oplus\mathds{Z}$ and $X$ satisfies the $K^2$-condition. Birational rigidity conjecture. Theorem 1.2 should extend to del Pezzo fibrations of degree $2$ with only singularities of type $\frac{1}{2}(1,1,1)$, and to del Pezzo fibrations of degree $1$ with only singularities of type $\frac{1}{2}(1,1,1)$ and $\frac{1}{3}(1,1,2)$. Such varieties should consequently be non-rational. This is described as a well-known and widely believed extension of the smooth birational-rigidity theorem; its resolution is not supplied here.