The Tango-curve characterization of Kawamata–Viehweg counterexamples on surfaces

Let $X$ be a normal projective surface. A counterexample to the Kawamata–Viehweg vanishing theorem means that the theorem fails for $X$. A Tango curve is a smooth projective curve $C$ with positive Tango invariant $n(C)>0$. A dominant rational map is a rational map whose image is dense.

Tango-curve characterization. If there is a counterexample to the Kawamata–Viehweg vanishing theorem on $X$, then there exists a dominant rational map

$$f:X\dashrightarrow C$$

to a smooth projective curve $C$ such that $C$ is a Tango curve.

This conjecture seeks a characterization of surface counterexamples in positive characteristic through fibrations or rational maps to Tango curves. The source presents it as a proposed converse to the known construction of counterexamples on geometrically ruled surfaces over Tango curves; its resolution status is not specified.