Bannister–Ostergård subdivision conjecture for stack number

Let $G$ be a graph, and let $H$ be a $(\leq 1)$-subdivision of $G$, obtained by subdividing each edge of $G$ at most once. Let $\operatorname{sn}(G)$ denote the stack number of $G$. Bannister–Ostergård subdivision conjecture. There is a function $f$ such that

$$\operatorname{sn}(G)\leq f(\operatorname{sn}(H)).$$

This conjecture asks whether stack number is bounded on graphs whose edges are subdivided at most once, in terms of the stack number of the original graph. The source states that the corresponding result is open and attributes the conjecture to Bannister and Ostergård.