Natural deformation conjecture for Jack-character structure coefficients

Let $cS_n$ be the symmetric group and let $cdelta$ be the Jack deformation parameter. The structure coefficients $g^\mu_{\pi,\sigma}$ are defined by

$$\operatorname{Ch}_\pi\operatorname{Ch}_\sigma=\sum_\mu g^\mu_{\pi,\sigma}(\delta)\operatorname{Ch}_\mu.$$

At $cdelta=0$, these coefficients describe multiplication of conjugacy-class indicators $A_\pi$ in the algebra of partial permutations. Natural-deformation conjecture. There exists some natural deformation of the symmetric group algebra $cmathbb{C}[\mathfrak{S}(n)]$ depending on $cdelta$ in which the structure coefficients for hypothetical conjugacy-class indicators are the coefficients $g^\mu_{\pi,\sigma}$ of Jack characters. Such a deformation would explain the positivity conjecture by extending the symmetric-group interpretation, but “natural” and the proposed algebra are not formally specified, and the source gives no resolution.