Is it possible that feedback vertex set problem has an $O(k^2/\log k)$ kernel？

(This question also suits for other similar natural $$\mathrm{NP}$$-hard problems)

I know that there is a $$4k^2$$ vertex kernel (and $$8k^2$$ edge kernel) by Thomasse [Thomasse09] for Feedback Vertex Set (FVS) problem.

Harnik and Naor [HN06] introduced a notion of compression with the goal of succinctly storing instances of computational problems.

As argued by Dell and van Melkebeek [DV14], for every real number $$\varepsilon > 0$$, FVS does not admits a polynomial compression with bitsize $$O(k^{2−\varepsilon})$$, unless $$\mathrm{NP} \subseteq \mathrm{coNP/poly}$$. It shows that obtaining kernels for the FVS problems with a strictly subquadratic number of edges is implausible (since $$O(k^{2−\varepsilon})$$ edges can be encoded in $$O(k^{2−\varepsilon}\log k)$$ bits).

However, it seems hard to understand, using solely the notion of the compositions as a tool, whether obtaining a kernel for FVS with a subquadratic number of vertices is possible or not.

Is it possible that FVS has an $$O(k^2/\log k)$$ kernel? In general, if there do exist an $$O(\mathrm{poly}(k))$$ kernel for a natural (graph) problem and it has a lower bound of size $$O(\mathrm{poly}(k))$$ bit. Is it still possible that this problem has an $$O(\mathrm{poly}(k)/\log k)$$ kernel? Do we have any evidences or obversations to believe or doubt it?