Vortex-cluster formula for the baryon number

Let $X=\mathbb{R}^3\cup\{\infty\}$ and let $U:X\to S^3$ be a map of degree $B$. Interpret $U$ as two vortices in $\psi_1$ and $\psi_2$ of $\boldsymbol{\psi}\in S^3$, with winding numbers $p_\ell$ and $q_\ell$ in each cluster. Vortex-cluster conjecture. The linking number $Q$ is

$$Q=\sum_{\ell}p_{\ell}q_{\ell},$$

and hence

$$B=Q=\sum_{\ell}p_{\ell}q_{\ell}.$$

This conjecture proposes an interpretation of degree-$B$ maps as pairs of vortices whose clusterwise winding numbers recover the baryon number through their linking. The source derives the relation from the stated corollary and theorem, but gives no resolution of the conjecture itself.