Non-negativity conjecture for deformed Verlinde expansion coefficients

Let ${\Greekmath 0115},{\Greekmath 0116} \in \mathcal{A}_{k,n}^{+}$ and ${\Greekmath 0117} \in \mathcal{\tilde{A}} _{k,n}^{+}$. Consider the expansion coefficients, or matrix elements,

$$\langle {\Greekmath 0115} | \boldsymbol{S}_{{\Greekmath 0117} ^{\prime }}^{\prime }|{\Greekmath 0116} \rangle.$$

Non-negativity conjecture. These expansion coefficients are always polynomials with non-negative coefficients. The statement is presented as an observation based on sample computations, and the supplied text gives no evidence that it has been resolved.