Cohomological dimension conjecture for the diagrammatic Clifford category

Let $d535a$ and $d535b$ be objects of the diagrammatic category $d538$. Write $d535ad53f$ and $d535bd53f$ for the corresponding dividing sets in the contact category, and call a dividing set $d535bd53f$ stackable over $d535ad53f$ when the corresponding contact-category morphism space is nonzero.

Cohomological dimension conjecture. For $d535a,d535b\in\mathcal{E}(\mathcal{CL})$,

$$\operatorname{dim}_{\mathbb{F}_2}H^*(\operatorname{Hom}_{\mathcal{CL}}(\mathbf{a},\mathbf{b}))=\begin{cases}1&\text{if }\overline{\mathbf{b}}\text{ is stackable over }\overline{\mathbf{a}},\\0&\text{otherwise.}\end{cases}$$

The conjecture predicts that the cohomology of every morphism complex in the diagrammatic category has dimension at most one, with nonvanishing exactly when the corresponding contact structures are related by a nonzero morphism. It is motivated by the uniqueness theorem for tight contact structures on the three-dimensional ball, but the authors state that the required bases and cohomology computations are not known.