The q≥4 sharpened Severi inequality conjecture

Let $S$ be a minimal surface of general type with Albanese dimension $2$ and irregularity $q(S)\ge 4$. The q≥4 conjecture. Then

$$K_S^2\ge 4p_g(S)-8.$$

Moreover, equality holds if and only if $S$ is a product of a curve of genus $2$ and a curve of genus at least $2$. This conjecture extends the proposed sharpened inequalities beyond the case $q(S)=3$; the source provides examples and presents the statement as open.