The topological-index conjecture for Lagrangian surfaces

Let $X$ be a Lagrangian surface, meaning a smooth projective surface whose holomorphic two-forms contain a nonzero decomposable element in the relevant kernel defining the Lagrangian condition. Let $\tau(X)$ denote its topological index. The topological-index conjecture. One has

$$\tau(X)\geq 0.$$

The preceding discussion proves non-negativity in several generalized Lagrangian cases under conditions on the divisorial base component, but this assertion concerns all Lagrangian surfaces.