Marino's LMOV integrality conjecture for composite invariants

Let $\mathcal{L}$ be a link with $L$ components. For tuples of partitions $\vec{A},\vec{B}\in\mathcal{P}^L$, let $h_{\vec{A}}(q,t)$ be the coefficients in the logarithm of the composite-invariant Chern–Simons partition function, and let $T_{AB}(x)$ be the character-theoretic matrix defined by $T_{AB}(x)=\sum_{\mu}\frac{\chi_A(C_\mu)\chi_B(C_\mu)}{z_\mu}p_\mu(x)$. Set $z=q-q^{-1}$.

Marino's conjecture. The reformulated quantities

$$\hat{h}_{\vec{B}}(q,t)=\sum_{\vec{A}}h_{\vec{A}}(q,t)T_{\vec{A}\vec{B}}(q^\rho)\in z^{-2}\mathbb{Z}[z^2,t^{\pm1}]$$

so there exist integers $N_{\vec{B},g,Q}$ such that

$$\hat{h}_{\vec{B}}(q,t)=\sum_{g\geq0}\sum_{Q\in\mathbb{Z}}N_{\vec{B},g,Q}z^{2g-2}t^Q.$$

The source says this conjecture was checked for many torus knots and links, but gives no resolution of the general claim.