The rank- cut-locus conjecture for free step-two Carnot groups

Let $\mathbb{G}_k$ be the free step-two Carnot group of rank $k$, identified with $\mathbb{R}^k\oplus\wedge^2\mathbb{R}^k$, and let $\mathrm{Cut}_0(\mathbb{G}_k)$ denote its cut locus from the identity. For $x\in\mathbb{R}^k$ and $Y\in\wedge^2\mathbb{R}^k$, write $Yx$ for the action of the skew-symmetric map associated with $Y$ on $x$. The cut-locus conjecture. The cut locus is

$$\mathrm{Cut}_0(\mathbb{G}_k)=P_k:=\{(x,Y)\mid Y=v\wedge w\neq 0,\,Yx=0\}.$$

This extends the characterization proved for ranks $k=2,3$ to arbitrary rank. The source presents it as one of two closely related conjectures; its resolution is not specified here.