Extreme black hole uniqueness conjecture for multiple degenerate black holes

A singular harmonic map is associated with a collection of $N>1$ distinct punctures on the $z$-axis, at positions $\mathbf{z}=(z_1,\ldots,z_N)$, with potential constants giving nonzero angular momenta $\mathcal{J}_i$ at each puncture. The tangent map parameters are denoted by $b_i$, for $i=1,\ldots,N$.

Extreme black hole uniqueness conjecture. At least one tangent map parameter must be nonzero:

$$\text{there exists }i\in\{1,\ldots,N\}\text{ such that }b_i\neq 0.$$

This is a PDE formulation of the claim that regular asymptotically flat multiple degenerate black hole solutions to the axisymmetric stationary vacuum Einstein equations do not exist. If all $b_i$ vanished, the associated stationary vacuum spacetime would have no conical singularities and would provide a counterexample to extreme black hole uniqueness.