Mariño's $SO/Sp$ Kauffman-invariant integrality conjecture

Let ${\mathcal K}$ be a knot, let ${\mathcal G}_R({\mathcal K})$ denote its Kauffman invariant in representation $R$, and let ${\mathcal H}_{(R_1,R_2)}({\mathcal K})$ denote the HOMFLY invariant colored by a composite representation. Define generating functions $Z_{H\overline H}$ and $Z_{\mathcal G}$ as in the source, define $g_R$ by

$$\log Z_{\mathcal G}(v)-\frac12\log Z_{H\overline H}(v)=\sum_{k\,\mathrm{odd}}\sum_R\frac1k g_R(q^k,\nu^k)s_R(v^k),$$

and set $\hat g_R(q,\nu)=\sum_S M_{RS}^{-1}g_S$, with $z=q-q^{-1}$. $SO/Sp$ Kauffman-invariant integrality conjecture. For every representation $R$,

$$\hat g_R(q,\nu)\in{\mathbb Z}[z,\nu^{\pm1}],$$

more precisely,

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

where $N^{c=1}_{R;g,Q}$ and $N^{c=2}_{R;g,Q}$ are integers, interpreted as BPS invariants for the non-orientable case. This is the non-orientable analogue of the HOMFLY integrality conjecture and implies congruence consequences for Kauffman invariants; its general status is not resolved in the source.