The primitive local-cohomology conjecture for character-variety cohomology

Let $M_{n,k}=H_L^n(O(k))^\vee$, and let $M_{n,k}^{\mathrm{shuffles}}\subset M_{n,k}$ be spanned by shuffle products $f*g$ with $f\in M_{n',k}$, $g\in M_{n-n',k}$, and $0<n'<n$. Define $M_{n,k}^{\mathrm{prim}}:=M_{n,k}/M_{n,k}^{\mathrm{shuffles}}$. Here $H^*(X)[\sum_{i=1}^nx_i,\sum_{i=1}^ny_i]$ denotes the indicated polynomial extension of the cohomology of the character variety. Primitive local-cohomology conjecture. There is an isomorphism of bigraded vector spaces

$$H^*(X)\left[\sum_{i=1}^nx_i,\sum_{i=1}^ny_i\right]\cong M_{n,k}^{\mathrm{prim}}.$$

The conjecture is motivated by computer experiments and identifies character-variety cohomology with primitive elements for the shuffle structure; the supplied text gives no resolution status.