Sharp parabolic estimates beyond outward-minimizing domains

Determine whether the optimal parabolic isoperimetric estimate for the space-time track of outward-minimizing level-set flow remains valid for more general smooth mean-convex domains.

Background

The paper proves an optimal parabolic isoperimetric estimate for the space-time track associated with an outward-minimizing level-set flow whose initial domain has smooth outward-minimizing boundary. The estimate has the shrinking round sphere as its unique equality case. The authors explicitly ask whether the same theorem extends to the broader class of smooth mean-convex domains, which need not be outward minimizing.