Strong Kato limit can be branching

We provide an example of a non-collapsed strong Kato limit that is branching, essentially branching, and satisfies neither the $\mathrm{C}\mathrm{D}(K,\mathrm{\infty })$ nor the $\mathrm{M}\mathrm{C}\mathrm{P}(K,N)$ conditions for any $K\in \mathbb{R}$ and $N\in [1,+\mathrm{\infty })$. In particular, this space is not a Ricci limit space. We also construct a compact non-collapsed strong Kato limit that cannot be obtained as Gromov-Hausdorff limit of closed Riemannian surfaces satisfying a uniform small $L^p$ bound à la Petersen–Wei.