  Kronheimer and Mrowka used gauge theory to define a functor $J^\sharp$ from a
category of webs in $\mathbb{R}^3$ to the category of finite-dimensional vector
spaces over the field of two elements. They also suggested a possible
combinatorial replacement $J^\flat$ for $J^\sharp$, which Khovanov and Robert
proved is well-defined on a subcategory of planar webs. We exhibit a
counterexample that shows the restriction of the functor $J^\sharp$ to the
subcategory of planar webs is not the same as $J^\flat$.
