The topological-index conjecture for minimal generalized Lagrangian surfaces

Let $X$ be a minimal generalized Lagrangian surface of general type. Thus, for a nonzero $w\in\ker\psi_2\subseteq H^0(X,\Omega_X^2)$ of rank $2$, let $V\subset H^0(X,\Omega_X^1)$ be the minimal-dimensional subspace with $w\in\bigwedge^2V$; assume that $V$ generically generates $\Omega_X^1$. Let $F_V$ be the divisorial part of the base scheme of $\psi_2(\bigwedge^2V)\subseteq H^0(X,\omega_X)$. It is contracted by $V$ if it is reduced and the restriction

$$V\otimes\mathcal O_X\longrightarrow\Omega_X^1\longrightarrow\Omega_{X|F_V}^1=\omega_{F_V}$$

vanishes. The topological-index conjecture. If $F_V$ is contracted by $V$, then the topological index satisfies

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

Earlier results establish non-negativity under additional hypotheses on $F_V$, including that it is zero or a reduced connected normal-crossings divisor contracted by $V$, while the conjecture asks whether those extra assumptions can be removed.