Let $X$ be a smooth complex projective surface. Is the Hurewicz image $h(\alpha)\in H\_2(X)$ of a homotopy class $\alpha\in\pi\_2(X)$ algebraic?



Is $\pi_2$ algebraic?
A slightly better variant of this question is to ask: is the Hurewitz image of $\pi\_{2}(X)$ in $H\_{2}(X)$ a sub Hodge structure? This is in fact an old question of Philippe Eyssidieux. In section 4.3 of [this](http://arxiv.org/abs/math/0107129) paper we proved that if the fundamental group of $X$ is nice enough (more precisely if $\pi\_{1}(X)$ is algebraically good) , then $\text{im}\left[ \pi\_{n}(X) \to H\_{n}(X)\right]$ is a sub Hodge structure for all $n$. 


