  Lazarev and Lieb showed that finitely many integrable functions from the unit
interval to $\mathbb{C}$ can be simultaneously annihilated in the $L^2$ inner
product by a smooth function to the unit circle. Here we answer a question of
Lazarev and Lieb proving a generalization of their result by lower bounding the
equivariant topology of the space of smooth circle-valued functions with a
certain $W^{1,1}$-norm bound. Our proof uses a relaxed notion of motion
planning algorithm that instead of contractibility yields a lower bound for the
$\mathbb{Z}/2$-coindex of a space.
