Finiteness conjecture for symmetric abstract 2-rigidity matroids

Let $n$ be a positive integer, let $\mathcal{R}_n$ denote the $2$-rigidity matroid on the edge set of the complete graph on $[n]$, and let $K_{3,3}$ be the complete bipartite graph with parts of size three. A matroid on this edge set is symmetric if it is invariant under relabelling the vertices, and it is an abstract $2$-rigidity matroid if it has the relevant $2$-rigidity rank properties.

Finiteness conjecture. There are only finitely many values of $n$ for which there exists a symmetric abstract $2$-rigidity matroid $\mathcal{M}_n\neq\mathcal{R}_n$ in which $K_{3,3}$ is independent.

The conjecture asks whether the exceptional distinction between symmetric $2$-rigidity matroids and the standard $2$-rigidity matroid can occur for arbitrarily large numbers of vertices. The paper gives an example at $n=6$ and reports no further examples at $n=7$, but does not resolve the question in general.