Parameter-independence and associativity conjecture for fusion products

Let $cmathfrak{g}$ be a semisimple complex Lie algebra, let $P^+$ be its dominant integral weights, and let $a_1,\ldots,a_sincmathbb C$ be pairwise distinct with clambda_1,\ldots,clambda_\sin P^+. The fusion product $V(clambda_1)_{a_1}castcdotscast V(clambda_s)_{a_s}$ is the associated graded module of the filtered tensor product of evaluation modules. Fusion-product independence and associativity conjecture. The fusion product is independent of the parameters $a_1,\ldots,a_s$, and the fusion product defined for any finite collection of cyclic modules is associative; in particular,

$$\left(V(\lambda_1)_{a_1} \ast V(\lambda_2)_{a_2}\right)_{b_1} \ast V(\lambda_3)_{a_3} \cong V(\lambda_1)_{a_1} \ast \left(V(\lambda_2)_{a_2} \ast V(\lambda_3)_{a_3}\right)_{b_1}.$$

These are foundational structural questions for fusion products and are stated as open in the supplied context.