The relaxed partition number conjecture

A path partition of a graph is a set of vertex-disjoint paths containing all vertices, and an isolated vertex counts as a path. For a graph $G$ with $n$ vertices, let $\pi_p(G)$ denote the minimum number of paths in a path partition, and let $d$ be its regular degree.

The relaxed partition number conjecture. For any $d$-regular graph $G$,

$$\pi_p(G)=O\left(\frac{n}{d+1}\right).$$

This is a weaker asymptotic version of the partition number conjecture. The supplied text gives no resolution status.