Reid's fundamental-group conjecture for surfaces of general type

Let $X$ be a minimal surface of general type, and let $\chi_{hol}$ denote its holomorphic Euler characteristic. Assume

$$K_X^2<4\chi_{hol}.$$

Reid's conjecture. The fundamental group $\pi_1(X)$ is either finite or commensurable with the fundamental group of a curve.

This conjecture is part of the geography theory of irregular surfaces. The source gives no resolution and discusses the related question of whether an aspherical complex surface can have negative signature.