The roots-of-unity MMR expansion conjecture for $F_K$

Let $\omega$ be a primitive $p$-th root of unity with $p\geq 1$, let $K$ be a nice knot, and let $F_K(x,q)$ be its $F_K$ series. Let $P_{K,\omega}^{(j)}(x)$ be the polynomials occurring in Habiro's roots-of-unity MMR expansion, and let $\Delta_K(x)$ be the Alexander polynomial. Roots-of-unity MMR conjecture for $F_K$. The series $F_K$ has an expansion at $q=\omega$ of the form

$$F_K(x,h+\omega)=\sum_{j\geq 0}\frac{P_{K,\omega}^{(j)}(x)}{\Delta_K(x^p)^{2j+1}}h^j,$$

where $P_{K,\omega}^{(j)}(x)\in\mathbb Z[\omega][x+x^{-1}]$ coincide with the polynomials in Habiro's conjecture. Both sides are understood as power series in $x$ and $h$. The conjecture is presented as an expected analogue and remains open.