Semisimple big-quotient conjecture for strong hyperbolicity

Let $X$ be a quasi-projective normal variety. A finitely generated group is semisimple if every abelian normal subgroup is finite. A quotient $\varrho:\pi_1(X)\twoheadrightarrow G$ is big in the sense used by the source if it satisfies the subsequently defined largeness condition. Semisimple big-quotient conjecture. If there exists such a quotient with $G$ semisimple and $\varrho$ big, then $X$ is strongly of log general type and pseudo-Picard hyperbolic. This proposes a group-theoretic characterization of strong hyperbolicity; the source gives no resolution.