Let $Y$ be a smooth compact Calabi-Yau three-fold (over $\mathbb C$, with $\pi_1(Y)=0$). Is it true that $c_2(Y)$ is Poincare dual to an effective curve? If not, what would be a simple counter-example?

Let $Y$ be a smooth compact Calabi-Yau three-fold (over $\mathbb C$, with $\pi_1(Y)=0$). Is it true that $c_2(Y)$ is Poincare dual to an effective curve? If not, can one construct a counter-example?

Note, that the answer to the same question for Fano $3$-folds is positive, as is stated here: Do all Fano threefolds have effective $c_2$? However, I don't understand if that answer applies to Calabi Yaus too.

Let $Y$ be a smooth Calabi-Yau three-fold (over $\mathbb C$). Is it true that $c_2(Y)$ is Poincare dual to an effective curve? If not, what would be a simple counter-example?

Let $Y$ be a smooth compact Calabi-Yau three-fold (over $\mathbb C$, with $\pi_1(Y)=0$). Is it true that $c_2(Y)$ is Poincare dual to an effective curve? If not, what would be a simple counter-example?