Integrality conjecture for reformulated colored Kauffman invariants of knots

Let ${\cal K}$ be a knot, let $R$ be a representation, and let $\hat h_R(t,\nu)$ and $\hat g_R(t,\nu)$ be the reformulated colored Kauffman invariants obtained by applying the inverse matrix $M^{-1}$ to the corresponding reformulated invariants. Set $z=t-t^{-1}$. Kauffman integrality conjecture for knots. There are integers $N^{c=0}_{R;g,Q}$, $N^{c=1}_{R;g,Q}$ and $N^{c=2}_{R;g,Q}$ such that

$$\hat h_R(t,\nu)\in z^{-1}{\mathbb Z}[z^2,\nu^{\pm1}],\qquad \hat g_R(t,\nu)\in{\mathbb Z}[z,\nu^{\pm1}],$$

with

$$\hat h_R(t,\nu)=z^{-1}\sum_{g\geq0}\sum_{Q\in{\mathbb Z}}N^{c=0}_{R;g,Q}z^{2g}\nu^Q,$$ $$\hat g_R(t,\nu)=\sum_{g\geq0}\sum_{Q\in{\mathbb Z}}\left(N^{c=1}_{R;g,Q}z^{2g}\nu^Q+N^{c=2}_{R;g,Q}z^{2g+1}\nu^Q\right).$$

This predicts an integral BPS-type expansion for the reformulated Kauffman invariants and is the Kauffman analogue of the colored HOMFLY integrality conjecture.