Skein-theoretic analogue of arithmetic boundary distributions

Determine the skein-theoretic analogue of the arithmetic boundary-condition distribution \(\mathcal{A}_G(X):\mathcal{A}_G(F)\to C\) and \(\mathcal{B}_{\check G}(\check X):\mathcal{B}_{\check G}(F)\to C\), including its categorical formulation for Hamiltonian \(G\)-spaces.

Background

The paper compares arithmetic Langlands boundary conditions with categorical distributions expected from skein theory. It proposes assignments from skein categories of boundary surfaces to vector spaces, but does not identify a definitive analogue of the arithmetic period or boundary distribution. The question is explicitly presented as a motivating unresolved problem.