Hawking Mass Compactness conjecture

Let $M_j^3$ be a sequence of three-dimensional oriented manifolds satisfying

$$\operatorname{Vol}(M_j)\le V_0,\qquad \operatorname{Vol}(\partial M_j)=A_0,\qquad \operatorname{Diam}(M_j)\le D_0.$$

Assume also

$$\operatorname{Scalar}_j\ge 0,\qquad H_{\partial M_j}\ge 0,\qquad m_H(\partial M_j)\le m_0,$$

and either that there are no closed interior minimal surfaces or that $\operatorname{MinA}(M_j)\ge A_1>0$. Hawking Mass Compactness conjecture. A subsequence should converge in the intrinsic flat sense, with

$$M_{j_k}\stackrel{\mathcal F}{\longrightarrow}M_\infty, \qquad {\mathbf M}(M_{j_k})\to {\mathbf M}(M_\infty),$$

and $M_\infty$ should satisfy the scalar-curvature and Hawking-mass bounds in some generalized sense. One might replace Hawking mass by another quasilocal mass. This is known in the rotationally symmetric setting when there are no closed interior minimal surfaces, but the general compactness statement remains open.