  The aim of this paper is to prove a conjecture made by T. Fukaya in 2008.
This conjecture concerns the exact value of the $\mathbb Z_2$-cup-length of the
Grassmann manifold $\widetilde G_{n,3}$ of oriented $3$-planes in $\mathbb
R^n$. Along the way, we calculate the heights of the Stiefel--Whitney classes
of the canonical vector bundle over $\widetilde G_{n,3}$.
