Cyclotomic expansion conjecture for colored $SU(n)$ invariants

Let $\mathcal{K}$ be a knot, let $n$ be fixed, and let $J_N^{SU(n)}(\mathcal{K};q)$ denote its colored $SU(n)$ invariant. For $N\geq 0$, define

$$C_{N+1,k}^{(n)}=\{N-(k-1)\}\{N-(k-2)\}\cdots \{N-1\}\{N\}\{N+n\}\{N+n+1\}\cdots \{N+n+(k-1)\}$$

for $k=1,\ldots,N$, with $C_{N+1,0}^{(n)}=1$.

Cyclotomic expansion conjecture. For any knot $\mathcal{K}$, there exist Laurent polynomials $H_k^{(n)}(\mathcal{K})\in\mathbb{Z}[q,q^{-1}]$, independent of $N$, such that

$$J_N^{SU(n)}(\mathcal{K};q)=\sum_{k=0}^{N}C_{N+1,k}^{(n)}H_k^{(n)}(\mathcal{K}).$$

In particular, $J_0^{SU(n)}(\mathcal{K};q)=H_0^{(n)}(\mathcal{K})=1$.

This is the cyclotomic expansion conjecture for colored $SU(n)$ invariants, motivated by congruence skein relations and Habiro's cyclotomic expansion for the colored Jones polynomial. The paper's abstract states that its results prove the formula for double twist knots, but the supplied text does not establish whether the conjecture in this general form has been resolved.