Horizontal joinability on $5$-dimensional $2$-step Carnot groups with a codimension $2$ horizontal distribution

For a $5$-dimensional $2$-step Carnot group $G_{3,2}$ with a codimension $2$ horizontal distribution, we prove that any two points $u,v\in G_{3,2}$ can be joined on it by a horizontal broken line consisting of at most three segments. A multi-dimensional generalization of this result is given.