The Constant Conjecture for cohomology rings of Hilbert schemes

Let $X$ be a quasi-projective surface. For each $n$, let $H^*(X^{[n]})$ be expressed in the Heisenberg monomial basis $(\mathfrak h_\mu(n))$. Constant Conjecture. The structure constants of the ring $H^*(X^{[n]})$ with respect to the Heisenberg monomial basis $(\mathfrak h_\mu(n))$ are independent of $n$. Consequently, one can construct a Farahat-Higman-type ring associated to $X$ encoding the cohomology rings $H^*(X^{[n]})$ for all $n$. The conjecture seeks a stable multiplication law for the cohomology of Hilbert schemes of points on quasi-projective surfaces, analogous to the Farahat-Higman ring for symmetric groups; the source does not state a resolution.