Generic nondegeneracy conjecture for short star-products

Let $\mathfrak g$ be the Lie algebra under consideration, let $s$ denote the degree-sign involution, and let ${{\mathbf A}}_\lambda$ be the corresponding filtered quantization parametrized by $\lambda$. Write $HH_0({{\mathbf A}}_\lambda,{{\mathbf A}}_\lambda s)$ for the zeroth Hochschild homology with coefficients twisted by $s$, and let $W$ be the relevant Weyl group. A short star-product is nondegenerate when its defining functional $T$ satisfies the nondegeneracy condition described in the paper. Generic nondegeneracy conjecture. (i) For Weil generic $\lambda$, a Weil generic element

$$T\in \bigl(HH_0({{\mathbf A}}_\lambda,{{\mathbf A}}_\lambda s)^s\bigr)^*$$

defines a nondegenerate short star-product corresponding to $g=s$. (ii) For Weil generic $\lambda$ such that $-\lambda\in W\lambda$, a Weil generic element

$$T\in \bigl(HH_0({{\mathbf A}}_\lambda,{{\mathbf A}}_\lambda s)^\sigma\bigr)^*$$

defines an even nondegenerate short star-product.