Minimal-volume conjecture for complex surfaces of general type

Let $M$ be a compact complex surface of general type, and let $\operatorname{Vol}_{K}(M)$ and $\operatorname{Vol}_{s}(M)$ denote its minimal volume defined using sectional curvature and scalar curvature, respectively. Minimal-volume conjecture.

$$\operatorname{Vol}_{K}(M)\geq 4\operatorname{Vol}_{s}(M),$$

with equality if and only if $M$ is complex hyperbolic. The proved estimate immediately before this conjecture has the weaker constant $\frac{9}{4}$; the conjectured sharp constant is motivated by the complex-hyperbolic case, and no resolution is stated here.