In dimension $m\geqslant 2,$ Example 3.32 describes a possible finite time singularity of graded, immersed Lagrangian MCF with $HF^{*}$ obstructed, after which one cannot continue the flow in graded Lagrangian MCF.

Such finite time singularities admit type II blow ups, as in Theorem 2.12, which are Lagrangian MCF translators from Example 2.16.

This is a generic singularity of Lagrangian MCF, that is, if Lagrangian MCF starting from $L^{0}$ develops such a singularity, then so does Lagrangian MCF starting from any sufficiently small Hamiltonian perturbation $\tilde{L}^{0}$ of $L^{0}$.

All this is possible for Lagrangians with arbitrarily small phase variation.