Manetti's equality-case conjecture for the Severi inequality

Let $S$ be a minimal smooth projective surface of maximal Albanese dimension, meaning that the image of its Albanese map is a surface. Write $q(S)$ for its irregularity, and let the Albanese map be $calpha:S\to\operatorname{Alb}(S)$.

Manetti's equality-case conjecture. The equality

$$K_S^2=4\chi(S)$$

holds if and only if $q(S)=2$ and the Albanese map has degree $2$.

This is the equality-case prediction for the Severi inequality and would imply the Keum–Naie surface characterization conjecture. The source presents it as a reasonable conjecture, while noting that the corresponding statement is known when the canonical bundle is ample.