Generalized Rains' positivity conjecture for specialized Koornwinder moments

Let $\operatorname{Par}_n$ denote the set of partitions with at most $n$ parts. A $(0,1)$-substitution map is an operator $\varphi$ on $\mathbb{Q}(\xi,\alpha,\beta,\gamma,\delta,q)$ that substitutes some, possibly none, of these parameters by $0$ or $1$, while $\varphi(\alpha\beta-q^i\gamma\delta)$ remains a non-constant polynomial for every integer $i\geq0$. Define the minimal numerator of $\varphi(M^Z_{\lambda,\mu})$ using the lowest-degree product of factors $\varphi(\alpha\beta-q^i\gamma\delta)$ that clears its denominator. Generalized Rains' conjecture. If $\varphi$ is a $(0,1)$-substitution map satisfying at least one of $\varphi(q)=1$, $\varphi(q)=0$, $\varphi(\alpha)=0$, $\varphi(\beta)=0$, $\varphi(\gamma)=0$, or $\varphi(\delta)=0$, then for all partitions $\lambda,\mu\in\operatorname{Par}_n$, the minimal numerator of $\varphi(M^Z_{\lambda,\mu})$ is a polynomial with nonnegative integer coefficients. The source notes that positivity fails for general pairs $(\lambda,\mu)$, while this conjecture asserts it under the listed specializations; its status is not specified.