Akbari et al.'s Caro–Wei bound for induced linear forests

All graphs are finite, simple, and undirected; for a graph $G$, let $V(G)$ be its vertex set and let $d(v)$ denote the degree of $v$. A linear forest is a forest in which every component is a path. If $G$ has minimum degree at least $2$, then Akbari et al.'s conjecture. $G$ has an induced linear forest with at least

$$\sum_{v\in V(G)} \frac{2}{d(v)+1}$$

vertices. This conjecture strengthens the $k=1$ case of the Alon–Kahn–Seymour bound, which guarantees an induced forest of this order but does not require its components to be paths. The conjecture is the motivating statement of the paper; the supplied text gives no evidence that it has been resolved.