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 $\mathrm{CD}(K,\infty)$ nor the $\mathrm{MCP}(K,N)$ conditions for any $K \in \mathbb{R}$ and $N \in [1,+\infty)$. 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 \`a la Petersen--Wei.