Polynomial degeneracy conjecture for graphs excluding subdivisions

For a graph $H$, call a graph $G$ $H$-subdivision-free if it has no subdivision of $H$ as an induced subgraph. Let $K_{\ell,\ell}$ denote the complete bipartite graph with $\ell$ vertices in each part, and let the degeneracy of $G$ be the least integer $d$ such that every induced subgraph of $G$ has a vertex of degree at most $d$.

Polynomial degeneracy conjecture for graphs excluding subdivisions. For every graph $H$, every $H$-subdivision-free graph $G$ that does not contain $K_{\ell,\ell}$ as a subgraph has degeneracy at most $f(H,\ell)$, where $f$ depends polynomially on $\ell$.

The analogous assertion for forbidden induced trees is known, with a bound polynomial in $\ell$, whereas the proposed extension to arbitrary graphs $H$ remains open.