Existence of equivariant bases for chord-diagram spaces

Let ${\cal C}^m_n$ denote the space of chord diagrams with $m$ distinguishable components and degree $n$, modulo the 4T-relation. The symmetric group on the components acts on this space. An equivariant basis is a basis preserved by this group action up to permutation. Equivariant-basis conjecture. There exists an equivariant basis for ${\cal C}^m_n$ for all $m$ and $n$. This would provide a basis compatible with the symmetric-group action relevant to links with indistinguishable components; the source does not give evidence that the conjecture has been resolved.