We construct nonunique solutions of the transport equation in the class $L^\infty$ in time and $L^r$ in space, for divergence free Sobolev vector fields from $W^{1,p}$. We achieve this by introducing two novel ideas: (1) in the construction, we interweave scaled copies of the vector field itself, and (2) asynchronous translation of cubes, which makes the construction heterogeneous in space. These new ideas allow us to prove nonuniqueness in the range of exponents going beyond what is available using the method of convex integration, and sharply match with the range of uniqueness of solutions from Bruè, Colombo, De Lellis ’21.