Enomoto and Ota's path-partition conjecture

Let $G$ be a graph of order $n$, let $k\geq 3$ be an integer, and let $n_{1},\dots,n_{k}$ be positive integers satisfying

$$\sum_{i=1}^{k}n_i=n.$$

Write $\sigma_2(G)$ for the minimum degree sum of two nonadjacent vertices. Enomoto and Ota's conjecture. If

$$\sigma_2(G)\geq n+k-1,$$

then for any $k$ distinct vertices $x_1,\dots,x_k$ in $G$, there exist vertex-disjoint paths $P_1,\dots,P_k$ such that $|P_i|=n_i$ and $P_i$ starts at $x_i$ for every $1\leq i\leq k$. The conjecture predicts a prescribed path partition of the entire vertex set under a degree-sum condition; the paper proves it when the order of $G$ is sufficiently large, while the remaining cases are not addressed here.