Ooguri–Vafa conjecture for framed links

Let $\mathcal{L}^{\vec{\tau}}$ be a framed $L$-component link with component framings $\vec{\tau}=(\tau_1,\ldots,\tau_L)$. For nonnegative integers $r_1,\ldots,r_L$, not all zero, let $k=\#\{r_i\mid r_i\neq 0, i=1,\ldots,L\}$, so that $1\leq k\leq L$, and let $\mathfrak{f}_{r_1,\ldots,r_L}(\mathcal{L}^{\vec{\tau}};q,a)$ be the coefficient functions obtained from the plethystic logarithm of the generating function of framed colored HOMFLYPT invariants. Ooguri–Vafa conjecture. There exist integers $N_{(r_1,\ldots,r_L),i,j}(\mathcal{L}^{\vec{\tau}})$, vanishing for sufficiently large $|i|$ and $|j|$, such that

$$\mathfrak{f}_{r_1,\ldots,r_L}(\mathcal{L}^{\vec{\tau}};q,a)=(q^{\frac{1}{2}}-q^{-\frac{1}{2}})^{k-2}\sum_{i,j\in\mathbb{Z}}N_{(r_1,\ldots,r_L),i,j}(\mathcal{L}^{\vec{\tau}})a^{\frac{i}{2}}q^{\frac{j}{2}}.$$

This is the proposed integrality and finite-support property of the framed-link BPS, or Ooguri–Vafa, invariants; the paper presents it as a generalization of the knot case, with no resolution supplied here.