Area-angular momentum inequality for multiple black holes

Let $(M,g,k)$ be a simply connected, axially symmetric, maximal initial data set with multiple ends, one $(M_{\text{end}}^{1})$ asymptotically flat and the others $(M_{\text{end}}^{i})$, $i=2,\ldots,I$ either asymptotically flat or asymptotically cylindrical. If the dominant energy condition is satisfied, and $P_i\eta^i=0$, then let $J$ be the sum of the angular momenta and let $A$ be the sum of the areas of the stable minimal surfaces enclosing each end $M_{\text{end}}^{i}$, $i=2,\ldots,I$. Multiple-black-hole area-angular momentum conjecture. One should have

$$\frac{A}{8\pi}\geq m^2-\sqrt{m^4-J^2}.$$

Moreover, equality should hold if and only if the initial data arise from an extreme Kerr spacetime. The inequality extends the corresponding single-black-hole area-angular momentum inequality to multiple black holes; unlike the charged case, additivity of area and subadditivity of total angular momentum support the proposed bound. The statement remains open in the source.