Bounded intrinsic diameter conjecture for mean curvature flow

Consider a mean curvature flow of surfaces approaching its first singular time, and let intrinsic diameter denote the diameter measured in the evolving surface's intrinsic metric. Bounded diameter conjecture. The intrinsic diameter stays uniformly bounded as one approaches the first singular time. This is motivated by an analogous conjecture for three-dimensional Ricci flow. In the setting discussed, it would follow from rigidity of the round cylindrical shrinker; a direct proof may require first understanding tangent-flow uniqueness.