One-parameter Albanese smoothness conjecture for nef-anticanonical varieties

Let $M$ be a normal projective variety with at most canonical Gorenstein singularities. Let $\varphi\colon M\rightarrow C$ be a fibration onto a smooth curve $C$ such that $-K_{M/C}$ is nef. Let $F$ be a general fibre. One-parameter Albanese smoothness conjecture. If $F$ is smooth, then $\varphi$ is smooth. If $F$ is smooth and simply connected, then $\varphi$ is locally trivial in the analytic topology. This problem is obtained in the paper as a reduction of the Albanese conjecture to fibrations over curves. Its resolution would therefore provide the corresponding one-dimensional base case for the original conjecture; no resolution is supplied in the stated passage.