Bannister–Oudot subdivision conjecture for stack-number

Let $G$ be a graph, let $k$ be an integer, and let $G'$ be any $(\leqslant k)$-subdivision of $G$, obtained by replacing each edge of $G$ by an internally disjoint path with at most $k$ internal vertices. Write $\operatorname{sn}(G)$ for the stack-number of $G$. Bannister–Oudot subdivision conjecture. There exists a function $f$ such that

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

The conjecture asserts that bounded-length subdivisions cannot reduce stack-number without a corresponding bound on the original graph's stack-number. The source states that a consequence of its construction resolves this conjecture, so the conjecture is solved.