Sparse-host induced subdivision extremal conjecture

Let $H$ be a graph, and let $\mathcal{D}_H$ be the property of not containing a subdivision of $H$ as an induced subgraph. For a graph $\Gamma$, write $\mathrm{ex}(\Gamma,\mathcal{D}_H)$ for the maximum number of edges in a subgraph of $\Gamma$ with this property. For $c>0$, say that $\Gamma$ is $(c,t)$-sparse when it satisfies the sparsity condition used in the paper. Sparse-host induced subdivision extremal conjecture. For every $c>0$ and a graph $H$, there exists $C>1$ such that, if $\Gamma$ is a $(c,t)$-sparse graph, then

$$\mathrm{ex}(\Gamma,\mathcal{D}_H)\le Ctn.$$

The motivation is a result showing that sufficiently dense graphs with no $K_{t,t}$ contain an induced subdivision of $H$; the stated sparse-host extremal estimate remains open.