Noether–Castelnuovo conjecture for irregular Gorenstein 3-folds of Albanese dimension one

Let $X$ be an irregular minimal Gorenstein 3-fold of general type with Albanese dimension one, meaning that the image of its Albanese map has dimension one. Let $K_X^3$ denote its canonical volume and let $\chi(\omega_X)$ be the Euler characteristic of its canonical sheaf.

Noether–Castelnuovo conjecture. One should have

$$K_X^3\ge 2\chi(\omega_X).$$

If the Albanese fiber has large volume, one should have the stronger inequality

$$K_X^3\ge 3\chi(\omega_X).$$

This is presented as a finer conjecture of Noether–Castelnuovo type after the paper proves stronger inequalities in related cases. The statement is not identified as resolved in the supplied text, so its status remains open.