Magnant–Wang–Yuan path-cover conjecture for graphs with prescribed degree bounds

For a graph $G$, let $\pi(G)$ denote the minimum number of vertex-disjoint paths needed to cover $V(G)$. Let $n=|V(G)|$, and write $\Delta(G)=\Delta$ and $\delta(G)=\delta$ for the maximum and minimum degrees.

Magnant–Wang–Yuan's conjecture. The path-cover number satisfies

$$\pi(G)\leq \max\left\{\frac{n}{\delta+1},\frac{(\Delta-\delta)n}{\Delta+\delta}\right\}.$$

The bound is tight for disjoint unions of $K_{\delta+1}$ and of $K_{\delta,\Delta}$. The conjecture is known when $\delta\leq 2$ and when $\Delta\geq 2\delta$, but remains open in general according to the source.