Manetti's geography inequality for irregular surfaces

Let $X$ be a minimal surface of general type and maximal Albanese dimension with irregularity $q\geq 4$. Manetti's geography conjecture. One should have

$$K_X^2\geq 4\chi(\mathcal O_X)+4(q-3).$$

This inequality concerns how the irregularity constrains the geography of surfaces of general type and maximal Albanese dimension. The source presents it as a conjecture of Manetti and states that it is proved there when $K_X^2\geq 36(q-2)$ or $\chi(\mathcal O_X)\geq 8(q-2)$; the supplied status therefore records it as resolved.