ISR conjecture for line graphs of simple bipartite graphs

Let $H$ be a simple bipartite graph, let $L(H)$ be its line graph, and let $V_1,\ldots,V_m$ be a partition of the edge set of $L(H)$. An independent system of representatives (ISR) is an independent set meeting each $V_i$ in at most one vertex. Bipartite line-graph ISR conjecture. If

$$|V_i|>\Delta(H)+1$$

for every $i$, then the system has an ISR. This proposes a stronger ISR property for line graphs of bipartite graphs than for general graphs; the source provides no evidence of a resolution.