Non-positive-curvature $C_2$ freeway basis conjecture

Fix a circle with $k$ points labeled single and $n$ points labeled double. Let $V_{1,0}$ and $V_{0,1}$ denote the corresponding fundamental representations, and let a switching class be an equivalence class under the switching moves. A $C_2$ freeway has non-positive curvature when every interior face has at least six sides.

Non-positive-curvature $C_2$ freeway basis conjecture. A collection of $C_2$ freeways of non-positive curvature, with one member in each switching class, constitutes a basis for

$$\operatorname{Inv}(V_{1,0}^{\otimes k}\otimes V_{0,1}^{\otimes n}).$$

Equivalently, the number of switching classes should equal the coefficient of $xy^2$ in

$$(x+y+x^{-1}+y^{-1})^k(1+xy+x^{-1}y+xy^{-1}+x^{-1}y^{-1})^n$$ $$\qquad\cdot(xy^2-x^{-1}y^2+x^{-2}y-x^{-2}y^{-1}+x^{-1}y^{-2}-xy^{-2}+x^2y^{-1}-x^2y).$$

Classical invariant theory gives spanning by non-positive-curvature freeways, while the switching classes account for the additional $C_2$ equivalences. The counting and basis assertion remain conjectural in the supplied text.