The flow-loop count equals the BPS quantum series for fibered links

Let $K$ be a fibered link, let $\Phi_{S^3 \setminus K}$ denote its $\mathrm{GL}_1$-skein-valued flow loop count, and let $\widehat{Z}_{S^3 \setminus K}$ denote the BPS $q$-series, normalized to begin with $1+\cdots$.

Flow-loop/BPS-series conjecture. For every fibered link $K$,

$$\Phi_{S^3 \setminus K}=\widehat{Z}_{S^3 \setminus K}.$$

The conjecture gives a precise formulation of the open Gromov–Witten invariants considered in the cited work. The BPS $q$-series is currently defined for braid-homogeneous links and, more generally, links admitting nice diagrams; when no independent definition is available, the assertion is interpreted as a resummation of the Melvin–Morton–Rozansky expansion of the colored Jones polynomials.