Coning principal-partition hypotheses in dimensions two and three

Determine whether hypotheses (a) and (b) in Theorem~\ref{thm:coning-k-circuits-strong} can be removed when coning a \(k\)-fold \(\mathcal{R}_d\)-circuit for \(d=2\) or \(d=3\).

Background

Theorem~\ref{thm:coning-k-circuits-strong} gives sufficient conditions ensuring that cone edges incident with two technicolour vertices lie in different parts of the principal partition after coning. A counterexample shows that these conditions cannot generally be removed for dimensions at least four.

The paper proves that the conditions can be removed in dimension one, but does not resolve whether the same strengthening holds in dimensions two and three.

References

We do not know whether these hypotheses are required when $d=2, 3$ but the following lemma shows that they are not required when $d=1$.

— $k$-fold circuits and coning in rigidity matroids  (2508.18838 - Hewetson et al., 26 Aug 2025) in Remark~\ref{rem:coning-counterexample}, Section 5 (The coning operation in rigidity matroids)