The integral-Tango characterization of Kodaira counterexamples

Let $Y$ be a smooth projective surface. A counterexample to the Kodaira vanishing theorem means that Kodaira vanishing fails for $Y$. A fibration is a morphism with one-dimensional fibres; a singular rational curve is a singular curve whose normalization is rational. A Tango curve of integral type is the type of Tango curve specified by the source's terminology.

Integral-Tango characterization. If there is a counterexample to the Kodaira vanishing theorem on $Y$, then there exists a fibration

$$g:Y\rightarrow C$$

to a smooth projective curve $C$ such that the general fibre of $g$ is a singular rational curve and $C$ is a Tango curve of integral type.

This is presented as the converse of a theorem constructing Kodaira-vanishing counterexamples from such fibrations, and as a statement similar to the broader Tango-curve conjecture. The source gives no evidence that it has been resolved.