The planar web dimension conjecture for instanton homology

Let $K$ be a web in $\R^{3}$, meaning an embedded trivalent graph, and let $\Jsharp(K)$ be the associated finite-dimensional vector space over the field $\F$ of two elements. A Tait coloring of $K$ is an assignment of the colors $\{1,2,3\}$ to its edges such that the three edges incident at every vertex have distinct colors. If $K$ lies in the plane, so that $K\subset\R^{2}\subset\R^{3}$, then the planar web dimension conjecture says that

$$\dim_{\F}\Jsharp(K)=\#\{\text{Tait colorings of }K\}.$$

The conjecture predicts that, for planar webs, the instanton-homological invariant is determined by the combinatorics of Tait colorings. The paper presents it as plausible evidence-based speculation; no resolution is supplied here.