Boundary-data conjecture for finite-energy monopole differences

Let ${\mathcal M}$ be the monopole moduli space, let ${\mathcal M}^{\infty}$ be the moduli space of boundary data at infinity, and let

$$\partial:{\mathcal M}\to{\mathcal M}^{\infty}$$

be the boundary map. For monopoles $(A,\Phi)$ and $(A',\Phi')$, equality of their boundary data means $\partial(A,\Phi)=\partial(A',\Phi')$. Boundary-data conjecture. If

$$\partial(A,\Phi)=\partial(A',\Phi'),$$

then there is a gauge transformation $g$ such that $A^g-A'$ and $\Phi^g-\Phi'$ are $L^2$. This would control the asymptotic difference between monopoles with identical boundary data and support finiteness of the induced $L^2$ metric.