The q=3 sharpened Severi inequality conjecture

Let $S$ be a minimal surface of general type with Albanese dimension $2$ and irregularity $q(S)=3$. The q=3 conjecture. Then

$$K_S^2\ge 4\chi(\mathcal O_S)+2=4p_g(S)-6,$$

and equality holds if and only if $S$ is the symmetric product of a curve of genus $3$. The examples preceding the conjecture motivate a sharper lower bound than the basic Severi inequality for irregularity $3$; the conjecture is presented as open in the source.