Non-positive-curvature $A_2$ freeway basis conjecture

Fix a circle with $k$ incoming points and $n$ outgoing points. Let $V_{1,0}$ and $V_{0,1}$ denote the corresponding fundamental representations, and let $\operatorname{Inv}$ denote the invariant subspace. An $A_2$ freeway is a crossingless $A_2$ diagram, and it has non-positive curvature when every interior face has at least six sides.

Non-positive-curvature $A_2$ freeway basis conjecture. The set of $A_2$ freeways of non-positive curvature having these points as endpoints is a basis for

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

Equivalently, the number of such freeways should equal the coefficient of $xy^2$ in

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

Classical invariant theory implies that freeways of non-positive curvature span the relevant space; the conjectural point is their linear independence, or equivalently the stated counting identity. The supplied text gives no general resolution.