Covering near-abelian groups by abelian subgroups: estimate $h(n)$ (Erdős #117)

Say a group $G$ has property $P(n)$ if every subset of more than $n$ elements of $G$ contains two distinct elements $x\ne y$ that commute ($xy=yx$). Let $h(n)$ be the least integer such that every group $G$ with property $P(n)$ can be covered by at most $h(n)$ abelian subgroups. Estimate $h(n)$ as sharply as possible — in particular determine its growth rate.

Acceptance. FULLY RESOLVES: determine the asymptotic growth of $h(n)$ — for instance prove $h(n)=c^{\,n+o(n)}$ for an explicit base $c$ (establishing $\lim_n h(n)^{1/n}=c$) with matching upper and lower bounds — via a complete proof (Lean/Coq preferred, else a full written proof). ADVANCES: improve either bound in Pyber's estimate $c_1^{\,n}<h(n)<c_2^{\,n}$ — a strictly larger explicit lower base or a strictly smaller explicit upper base than the best stated in the background — with proof; or determine $h(n)$ exactly for small $n$ via a verified construction and matching lower-bound argument. Deliver the proof of the improved bound (with the explicit constant), or the exact small-$n$ values together with their proofs.