Existence of equivariant bases for link chord-diagram spaces

Let ${\cal A}^m_n$ denote the corresponding degree-$n$ chord-diagram space for links with $m$ components, with the induced symmetric-group action on components. An equivariant basis is a basis compatible with this action. Equivariant-basis conjecture. There exists and equivariant basis for ${\cal A}^m_n$ for all $m$ and $n$. This is presented as a general conjecture following the claim for ${\cal C}^m_n$, extending the expected equivariant-basis structure to the link chord-diagram spaces; the source gives no resolution.