Hriňáková–Knor–Škrekovski conjecture on extremal iterated-line-graph Wiener ratios

Let $G$ be a graph on $n$ vertices. The Wiener index $W(G)$ is the sum of distances over all unordered pairs of vertices. Let $L(G)$ be the line graph of $G$, define $L^0(G)=G$ and $L^k(G)=L(L^{k-1}(G))$ for $k\geq 1$, and set

$$R_k(G)=\frac{W(L^k(G))}{W(G)}.$$

Hriňáková–Knor–Škrekovski conjecture. For large $n$ and $k\geq 2$, among all graphs $G$ on $n$ vertices, $R_k(G)$ attains its maximum at $G=K_n$ and its minimum at $G=P_n$.

The minimum for $k=1$ is known to be attained by the star, while for trees and $k\geq 3$ the path is known to attain the minimum. The conjecture concerns the remaining extremal cases and the simultaneous maximum and minimum over all graphs.