Planar maximal 3-degenerate graphs are good

Let $d\geq 1$, and let $H$ be a graph of order $d+1$. Say that $H$ is good if every graph of minimum degree at least $d$ contains a subdivision of $H$.

Planar maximal 3-degenerate graph conjecture. Every planar maximal 3-degenerate graph is good.

The paper notes that non-planar 3-degenerate graphs of order $6$ are not all good, but reports no planar counterexample. This statement is presented as a problem and remains open in the source.