Is something known about the higher homotopy groups of Calabi-Yau threefolds? For example, one of the easiest CYs is the quintic, defined as the anticanonical divisor in $\mathbb{CP}_4$. What are its homotopy groups?

In particular, here a similar question was asked for K3. What are the higher homotopy groups of a K3 suface? In the first answer, an interesting theorem is cited. It says that the homotopy group of any simply-connected closed 4-manifold $M$ are determined by the second Betti number of $M$. Call it $k$. Then, if $k\geq 1$ and $j\geq 3$ then $\pi_j(M)=\pi_j(\#^{k-1}S^2\times S^3)$. Does a similar result exist for 6-manifolds?