Let $X$ be an algebraic surface of general type. Assume $K_X$ is an integer multiple of another class $A$, and the class $A$ can be represented by a symplectic submanifold $S$ of $X$ with non-negative self-intersection. Is it true that $\pi_{1}(S)$ surjects into $\pi_{1}(X)$? Note that I don't assume $S$ is a complex submanifold.
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