Li, Wu, Meng and Ma's $k$-tree connectivity conjecture for line graphs

Let $G$ be a connected graph with at least $k$ vertices and at least $k$ edges, and let $L(G)$ be its line graph. For a set $S\subseteq V(G)$ with $|S|\geq 2$, let $\lambda_G(S)$ be the maximum number of edge-disjoint $S$-Steiner trees in $G$, and define $\lambda_k(G)$ as the minimum of $\lambda_G(S)$ over all $k$-subsets $S$ of $V(G)$. Similarly, let $\kappa_k(L(G))$ denote the minimum, over all $k$-subsets of $V(L(G))$, of the maximum number of internally disjoint Steiner trees joining that subset. Li, Wu, Meng and Ma's conjecture. For every integer $k\geq 2$, if $G$ is connected and has at least $k$ vertices and at least $k$ edges, then

$$\kappa_k(L(G))\geq \lambda_k(G).$$

The conjecture would give a sharp relationship between generalized vertex-connectivity of a line graph and generalized edge-connectivity of the original graph. It is known for $k=2$, $k=3$, and $k=4$, while the general case is presented as an open conjecture.