The asymmetric simple exclusion process (ASEP) is an important particle model with deep connections to orthogonal polynomials. Motivated by this connection, Corteel and Williams introduced the Koornwinder moments $M^{Z}_λ$ at $ t=q $, which generalize the moments of Askey--Wilson polynomials. They showed that the partition function of the two-species ASEP is equal to $M^{Z}_λ$ for a one-row partition $ λ$. In this paper, we investigate a conjecture of Rains on the positivity of the minimal numerator of the Koornwinder moment $M^{Z}_λ$. We derive the first explicit formula for this moment, thereby obtaining a precise formulation of the conjecture by determining the minimal denominator of $M^{Z}_λ$. We also propose a generalization of the conjecture for the more general Koornwinder moments $M^{Z}_{λ,μ}$ indexed by two partitions at special parameter values. We prove the generalized Rains' conjecture in two special cases: $(ξ,q)=(1,0)$ and $(ξ,q)=(1,1)$. For $(ξ,q)=(1,0)$, we construct a lattice path model and obtain a combinatorial formula for $M^{Z}_{λ,μ}$ in terms of non-intersecting lattice paths. For $(ξ,q)=(1,1)$, we establish an explicit product formula for $M^{Z}_{λ,μ}$ and give a combinatorial interpretation using lecture hall tableaux.