Determine validity of the 13-point incidence theorem over skew fields

Determine whether the 13-point incidence theorem of Theorem 1.7, asserting the collinearity of p_{11}, p_{12}, and p_{13} under its specified incidence conditions, holds over all skew fields.

Background

Theorem 1.7 establishes the 13-point incidence theorem over every commutative field while exhibiting a commutative-ring counterexample showing that it is not an absolute incidence theorem. The paper notes that incidence theorems over skew fields have a distinct theory and that, unlike the field case, it is undecidable in general to determine whether an incidence theorem holds over all skew fields. The authors leave unresolved whether this particular incidence theorem remains valid in that setting.