Is the sum of reciprocals of cycle lengths minimised by complete bipartite graphs? (Erdős #65)

Let $G$ be a graph with $n$ vertices and $kn$ edges, and let $a_1<a_2<\cdots$ be the distinct lengths of cycles in $G$. Is it true that $$\sum\frac{1}{a_i}\gg \log k?$$ Is the sum $\sum\frac{1}{a_i}$ minimised when $G$ is a complete bipartite graph? (The first question is settled: Gyárfás, Komlós, and Szemerédi proved $\sum 1/a_i\gg\log k$ — see background. The open problem is the second, extremal question: among graphs with $n$ vertices and $kn$ edges, is the complete bipartite graph the minimiser of $\sum 1/a_i$?)

Acceptance. FULLY RESOLVES: a proof that, for all $n$ and $k$ in a clearly stated regime (all $k\geq 2$, or all sufficiently large $k$), every graph with $n$ vertices and $kn$ edges has $\sum 1/a_i$ at least the value attained by the complete bipartite graph with the same number of vertices and at least as many edges — i.e. complete bipartite graphs minimise the sum, exactly or up to an explicitly stated asymptotic precision; OR a disproof: an explicit family of graphs (with proofs of their edge counts and full cycle spectra) whose reciprocal cycle-length sum is strictly below the complete bipartite value. Machine-checkable proofs (Lean/Coq) preferred, else a complete written proof. Any submission must state its relationship to the forthcoming Milojević–Montgomery–Pokrovskiy–Sudakov work flagged in the background and go beyond whatever part of it is public. ADVANCES: improve the lower-bound constant strictly beyond the $(\tfrac{1}{2}-o(1))\log k$ of Liu–Montgomery as stated in the background (e.g. matching the bipartite value up to an additive constant); prove the extremal statement within a restricted graph class (e.g. among bipartite graphs); or a reproducible exhaustive computation determining the exact minimisers of $\sum 1/a_i$ over all graphs with $n$ vertices and $m$ edges for all $(n,m)$ up to a stated bound, with code and an exhaustiveness certificate (new data — no such tables are recorded on the source page). Deliver the proof file, or the enumeration code plus verified extremal tables.