Brunella's mixed Nevanlinna-current conjecture

Let $X$ be a compact projective surface of general type. A Nevanlinna current associated with an entire curve $f:\mathbb{C}\rightarrow X$ has a Siu decomposition

$$\mathsf{T}=\mathsf{T}_{\text{alg}}+\mathsf{T}_{\text{diff}},$$

with singular part $\mathsf{T}_{\text{alg}}$ and diffuse part $\mathsf{T}_{\text{diff}}$.

Brunella's mixed Nevanlinna-current conjecture. Some entire curve shall produce a Nevanlinna current $\mathsf{T}=\mathsf{T}_{\text{alg}}+\mathsf{T}_{\text{diff}}$ with both nontrivial singular part $\mathsf{T}_{\text{alg}}\neq 0$ and nontrivial diffuse part $\mathsf{T}_{\text{diff}}\neq 0$.

This is the opposite alternative to Brunella's zero–one conjecture and is explicitly described in the source as still open. The source compares the dichotomy with a zero–one law and suggests that an Oka-principle construction may yield a counterexample to the first conjecture.