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

The combinatorial and gauge-theoretic foam evaluation functors are not the same