Knop–Sahi positivity conjecture for normalized interpolation polynomials

Let $\delta=(n-1,\ldots,1,0)$, let $r$ be a parameter with $\alpha=1/r$, and let $P_{\lambda }^{r\delta }$ be the interpolation polynomial indexed by a partition $\lambda$. Define

$$J_{\lambda }^{r\delta}=(-1)^{|\lambda|}c_{\lambda}(\alpha)P_{\lambda}^{r\delta}(-x),$$

where $c_{\lambda}(\alpha)$ is the Jack normalization factor, and write

$$J_{\lambda }^{r\delta}=\sum_{\mu}\alpha^{|\mu|-|\lambda|}a_{\lambda,\mu}(\alpha)m_{\mu}.$$

Knop–Sahi positivity conjecture. The coefficients satisfy

$$a_{\lambda,\mu}(\alpha)\in\mathbb{N}_0[\alpha].$$

This conjecture generalizes Macdonald's positivity conjecture from Jack polynomials to interpolation polynomials. The paper's abstract states that the Knop–Sahi conjecture is proved, so the conjecture is now a theorem.