Ilmanen's mean-convex neighborhood conjecture

Consider a cylindrical singularity of mean curvature flow. A neighborhood is mean-convex when the flow in that neighborhood has the corresponding one-sided mean-convex orientation. Ilmanen's mean-convex neighborhood conjecture. Cylindrical singularities have a mean-convex neighborhood. The conjecture is known for neck singularities, meaning cylinders with only one $\mathbb{R}$-factor, but remains open in general; the source describes a possible route to proving it in dimension four and eventually all dimensions.