The GM series surgery conjecture for links

Let $L\subset S^3$ have components $L_1,\ldots,L_l$, let $S^3_{p_1,\ldots,p_l}(L)$ be the result of $(p_1,\ldots,p_l)$-surgery, and let $\mathrm{B}$ be its linking matrix, with diagonal entries $p_i$ and off-diagonal entries $lk(i,j)$. Define $\mathcal{L}^{b}_{\mathrm{B}}$ by

$$\mathcal{L}^{b}_{\mathrm{B}}:x^u\mapsto\begin{cases}q^{-(u,\mathrm{B}^{-1}u)}&u\in b+\mathrm{B}\mathbb{Z}^l,\\0&\text{otherwise}. \end{cases}$$

The link surgery conjecture. Whenever the right-hand side makes sense, for some $\epsilon\in\{\pm1\}$ and $d\in\mathbb{Q}$,

$$\hat Z_b(S^3_{p_1,\ldots,p_l}(L))=\epsilon q^d\mathcal{L}^{b}_{\mathrm{B}}\left[\prod_{i=1}^l(x_i^{\frac12}-x_i^{-\frac12})F_L(x_1,\ldots,x_l,q)\right].$$

This is the link extension of the surgery formula and is presented as a conjectural, experimentally supported identity.