Nonsmooth subvariety conjecture for higher cylindrical singular strata

Let $n\geq 3$ and $k\geq 2$. For a mean curvature flow $\mathbf{M}$ in $\mathbb{R}^{n+1}$, let $\mathcal{S}_{k}(\mathbf{M})_+$ denote its positive part of the $k$-th cylindrical singular stratum. Higher-stratum nonsmoothness conjecture. There exists a mean curvature flow $\mathbf{M}$ in $\mathbb{R}^{n+1}$ such that $\mathcal{S}_{k}(\mathbf{M})_+$ is not a $k$-dimensional submanifold but is a nonsmooth subvariety of dimension at most $k-1$. The claim would show that the containment of $\mathcal{S}_{k}(\mathbf{M})_+$ in a $k$-dimensional submanifold cannot generally be improved to equality; the source suggests examples analogous to intersecting curves and leaves the existence of such flows open.