Spectral convergence consequence of the Schoen–Yau zero mass stability conjecture

Let $\Omega_j(R)$ be the isoperimetric regions associated with a sequence satisfying the hypotheses of the zero mass stability conjecture, and let $B_0(R)\subset{\mathbb E}^3$ be the Euclidean ball of radius $R$. Spectral convergence consequence. Under the hypotheses of that conjecture, the eigenvalues of the Laplacian of $\Omega_j(R)$ semi-converge to the eigenvalues of the Laplacian of $B_0(R)$ in the sense described by Portegies.

This is proposed as a consequence of volume-preserving intrinsic flat convergence; the source does not state that it has been proved, so it remains open.