The conjecture on simple fibrations for canonical threefolds on the Noether line

A canonical threefold on the Noether line is a canonical threefold whose canonical volume and geometric genus satisfy

$$K^3=\frac{4}{3}p_g-\frac{10}{3}.$$

A simple fibration in $(1,2)$-surfaces means the type of fibration specified in the source, with fibres that are $(1,2)$-surfaces, over the projective line $\mathbb{P}^1$. The simple-fibration conjecture. Every canonical threefold on the Noether line with $p_g$ sufficiently large birationally admits a simple fibration in $(1,2)$-surfaces over $\mathbb{P}^1$.

The conjecture is the starting point for the paper's proof of the complete description of the relevant moduli spaces; the supplied text does not state whether it has been proved in full or remains open.