The asymptotic t-broom conjecture for distance graphs

Let $G$ be a graph, let $G_k$ denote its distance-$k$ graph, and let $e(G_k)$ be the number of edges of $G_k$. Two graphs are $k$-isomorphic when their distance-$k$ graphs are isomorphic; a $t$-broom is the graph construction described in the paper, with $t$ broom branches.

The asymptotic t-broom conjecture. For every integer $k\geq 3$, there exists a function $h=h(k)$ such that, whenever $n\geq h(k)$, the maximum of $e(G_k)$ over all graphs $G$ with $|G|=n$ is attained when $G$ is $k$-isomorphic to a $t$-broom for some $t$.

The conjecture is motivated by the known extremal result for trees, while the paper notes that small cases can have better constructions, including the $7$-cycle when $(n,k)=(7,3)$. The authors state that they believe the conjecture but cannot prove it.