The codegree decomposition conjecture for point configurations

Let $\boldsymbol{\mathbf{A}}$ be a point configuration of dimension $d$, and let $\operatorname{deg}(\boldsymbol{\mathbf{A}})$ denote its degree. A codegree decomposition consists of disjoint nonempty subsets of $\boldsymbol{\mathbf{A}}$ satisfying the face and codegree additivity conditions, and its length is the number of factors.

Codegree decomposition conjecture. If

$$d>2\operatorname{deg}(\boldsymbol{\mathbf{A}}),$$

then $\boldsymbol{\mathbf{A}}$ admits a codegree decomposition of length at least

$$d+1-2\operatorname{deg}(\boldsymbol{\mathbf{A}}).$$

This is a strengthening of the weak Cayley configuration conjecture: a codegree decomposition implies a weak Cayley configuration of the same length. The source presents this as the main structural conjecture, with special cases proved.