Root-of-unity rationality conjecture for the unified knot invariant

Let $K$ be a knot and let $\zeta$ be a root of unity of order $r$. Let

$$\gamma _\zeta\colon \Lambda \to \mathbb{Z}[\zeta ][[\alpha ^2,q-\zeta ]]$$

be induced by $\mathbb{Z}[q,q^{-1},\alpha ^2]\subset\mathbb{Z}[\zeta ][q,q^{-1},\alpha ^2]$. Define $P_{K,\zeta,k}(t)\in\mathbb{Z}[\zeta ][[\alpha ^2]]$ by

$$\gamma _\zeta(J_K)=\sum_{k\ge 0}\frac{P_{K,\zeta,k}(t)}{\Delta _K(t^r)^{2k+1}}(q-\zeta )^k.$$

Root-of-unity rationality conjecture. For every $k\ge 0$, one has $P_{K,\zeta,k}(t)\in\mathbb{Z}[\zeta ][t+t^{-1}]$. The source presents this as implied by the preceding cyclotomic rationality conjecture; the case $\zeta =1$ is Rozansky's rationality theorem.