Path-factor existence conjecture for graphs without small odd paths

Let $k\geq 3$, and let $G$ be a graph. For a graph $H$, let $\mathcal{C}(H)$ be the set of components of $H$; for $i\geq 1$, let $\mathcal{C}_i(H)=\{C\in\mathcal{C}(H)\mid |V(C)|=i\}$ and let $c_i(H)=|\mathcal{C}_i(H)|$. A $\{P_2,P_{2k+1}\}$-factor is a spanning subgraph whose components are copies of $P_2$ or $P_{2k+1}$. Path-factor existence conjecture. If

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

for all $X\subseteq V(G)$, then $G$ has a $\{P_2,P_{2k+1}\}$-factor. The conjecture extends the corresponding sufficient conditions known for $k=1$ and $k=2$. For $k\geq 3$ divisible by $3$, examples show that the analogous bound with an additional constant term cannot guarantee such a factor, while the conjecture proposes that removing this constant term suffices in every case.