Jeffs's convexity conjecture for neural codes with four maximal codewords

Let $\mathcal{C}$ be a neural code, meaning a collection of subsets of a neuron set, and call a codeword maximal if it is not properly contained in another codeword. A code is convex if it can be realized as the code of a collection of convex open sets in Euclidean space. A local obstruction and a wheel are the combinatorial obstructions described in the source.

Jeffs's conjecture. If $\mathcal{C}$ has up to four maximal codewords, then $\mathcal{C}$ is convex if and only if $\mathcal{C}$ has no local obstructions and no wheels.

This conjecture extends the known characterization for codes with at most six neurons and four maximal codewords, where convexity is equivalent to avoiding local obstructions and wheels. Its status is not specified in the source.