  We construct smooth mean curvature flows with surgery that approximate weak
mean curvature flows with only spherical and neck-pinch singularities. This is
achieved by combining the recent work of Choi-Haslhofer-Hershkovits, and
Choi-Haslhofer-Hershkovits-White, establishing canonical neighbourhoods of such
singularities, with suitable barriers to flows with surgery. A limiting
argument is then used to control these approximating flows. We conclude by
improving the entropy bound on the low-entropy Schoenflies conjecture.
