Rains's positivity conjecture for Koornwinder moments

Let $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_m)$ be a partition, and let $K_{\lambda}(\xi)$ denote the Koornwinder moment defined by

$$K_{\lambda}(\xi)=\frac{\det(Z_{\lambda_i+m-i+m-j}(\xi))_{i,j=1}^m}{\det(Z_{2m-i-j}(\xi))_{i,j=1}^m},$$

where $Z_N$ is the ASEP partition function and $\xi$ denotes its parameters. Rains's conjecture. The Koornwinder moment $K_{\lambda}(\xi)$ is a polynomial in $\alpha,\beta,\gamma,\delta,q,\xi$ with positive coefficients, up to a normalizing factor. The conjecture is motivated by the positivity of the ASEP partition function, which is a polynomial with positive coefficients up to normalization; the paper establishes connections between these moments and the two-species ASEP but does not resolve the general positivity claim.