Horizontal joinability in canonical 3-step Carnot groups with corank 2 horizontal distributions

We prove that on each 2-step Carnot group with a corank 1 horizontal distribution two arbitrary points can be joined with a horizontal broken line consisting of at most 3 segments, while on every canonical 3-step Carnot group $\Bbb G$ with a corank 2 horizontal distribution two arbitrary points can be joined with a horizontal broken line consisting of at most 7 segments. We show that two arbitrary points in the center of $\Bbb G$ are joined by infinitely many horizontal broken lines with 4 segments. Here by a segment of a horizontal broken line we mean a segment of an integral line of some left-invariant horizontal vector field that is a linear combination of left-invariant horizontal basis vector fields of the Carnot group.