Conjecture on decomposable derived categories of elliptic fibrations

Let $X$ be a minimal surface of Kodaira dimension $1$ with an elliptic fibration

$$\pi:X\longrightarrow \mathbb{P}^1.$$

Suppose that $P_g(X)=0$ and $q(X)=1$. Elliptic-fibration decomposability conjecture. Then $D(X)$ admits some nontrivial semi-orthogonal decompositions. The conjecture concerns the remaining case in the paper's analysis of elliptic fibrations with $P_g(X)=0$ and positive irregularity; no resolution is indicated here.