Cup-length of oriented Grassmann manifolds via Gr\"obner bases

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 $\widetilde G_{n,3}$.