The Rational Conjecture for non-Gorenstein normal surface singularities

Let $(X,E)\rightarrow (V,0)$ be the minimal good resolution (MGR) of a complex normal surface singularity, and define $S_X=(\Omega^1_X(\operatorname{log} E))^*$. Assume that $(V,0)$ is not Gorenstein. The Rational Conjecture. One has

$$h^1(\mathcal O_X)-h^1(S_X)+h^1(-(K_X+E))\geq 0,$$

with equality if and only if $(V,0)$ is quasihomogeneous. This formulation relates the geometric genus, equisingular deformation dimension, and second plurigenus. The source presents it as the broader conjectural inequality from which the rational-singularity formulation is a special case; its general status is not resolved in the supplied text.