Severi equality characterization for irregular surfaces

Let $X$ be a minimal surface of general type and maximal Albanese dimension. The Severi equality characterization states that

$$K_X^2=4\chi(\mathcal O_X)$$

if and only if the canonical model of $X$ is a flat double cover of an abelian surface branched over an ample divisor with at worst simple singularities. Here a branch divisor has at worst simple singularities when the multiplicities of the singularities appearing in the process of the canonical resolution are at most three. The characterization was confirmed under the assumption that $K_X$ is ample, while the general statement is presented in the source as a conjecture.