The level set flow of a hypersurface in $\mathbb R^4$ of low entropy does not disconnect

We show that if $\Sigma\subset \mathbb R^4$ is a closed, connected hypersurface with entropy $\lambda(\Sigma)\leq \lambda(\mathbb{S}^2\times \mathbb R)$, then the level set flow of $\Sigma$ never disconnects. We also obtain a sharp version of the forward clearing out lemma for non-fattening flows in $\mathbb R^4$ of low entropy.