Fattening dichotomy conjecture for mean curvature flow translators

Let $T[\Sigma_0]$ denote the level-set flow associated with the translator $\Sigma_0$, and suppose that $T[\Sigma_0]$ fattens. A complete translator is called unstable or \mathcal{I}-stable according to the stability notion used in the source. Fattening dichotomy conjecture. If $T[\Sigma_0]$ fattens, then either there exists an unstable complete translator in the interior of $T[\Sigma_0]$, or $T[\Sigma_0]$ admits a foliation by complete $\mathcal{I}$-stable translators. The conjecture describes the proposed alternatives when fattening occurs; the source gives no resolution, so both alternatives remain open in general.