Bannister–Ojakian's shallow topological-minor conjecture for stack-number

Let $G$ and $H$ be graphs, and let $k\geqslant 0$ be a half-integer. A graph $H$ is a $k$-shallow topological minor of $G$ if a subgraph of $G$ is isomorphic to a subdivision of $H$ in which each edge is replaced by a path with at most $2k$ internal vertices. Write $\operatorname{sn}(G)$ for the stack-number of $G$. Bannister–Ojakian's conjecture. There exists a function $f$ such that for every graph $G$ and half-integer $k\geqslant 0$, if $H$ is any $k$-shallow topological minor of $G$, then

$$\operatorname{sn}(H)\leqslant f(\operatorname{sn}(G),k).$$

The conjecture proposed that stack-number is well-behaved under shallow topological minors. It was disproved by Dujmović, Eppstein, Hickingbotham, Morin, and Wood (2021).