Rozansky-type cyclotomic rationality conjecture for knots

Let $K$ be a knot and let $d\colon \mathbb{N}\to\{0,1,2,\ldots\}$ vanish except at finitely many elements. Define

$$\Delta ^d_K(t)=\prod_{r\in\mathbb{N},\,d(r)>0}\Delta _K(t^r)^{2d(r)-1}\in\mathbb{Z}[t+t^{-1}].$$

Let $\Lambda$ be the cyclotomic completion containing the unified knot invariant $J_K$, and let $\Phi _d(q)$ be the corresponding cyclotomic ideal element. Rozansky-type rationality conjecture. One has

$$\Delta ^d_K(t)J_K\in \mathbb{Z}[t+t^{-1},q,q^{-1}]+\Phi _d(q)\Lambda.$$

The source says this conjecture is supported by computer calculations and that it generalizes Rozansky's rationality theorem; it also implies the subsequent root-of-unity rationality conjecture.