Egawa–Furuya conjecture on path-factors and odd components

Let $k\geq 3$ be an integer, let $G$ be a graph, and for a vertex set $X\subseteq V(G)$ let $c_{2j+1}(G-X)$ denote the number of components of $G-X$ having order $2j+1$. A {$P_{2},P_{2k+1}\}$-factor is a spanning subgraph whose components are isomorphic to $P_{2}$ or $P_{2k+1}$. Egawa–Furuya's conjecture. If

$$\sum_{0\leq j\leq k-1}c_{2j+1}(G-X)\leq \frac{4k+6}{8k+3}|X|$$

for all $X\subseteq V(G)$, then $G$ has a $\{P_{2},P_{2k+1}\}$-factor. This conjecture was proposed after examples showing that, for $k\geq 3$ with $k\equiv 0\pmod 3$, the corresponding bound with an additional constant term cannot generally be improved away; the paper proves the weaker sufficient bound $\frac{5}{6k^{2}}|X|$, but does not resolve the conjectured coefficient.