Partitions of a graph into paths with prescribed endvertices and lengths

For a graph G, let σ₂(G) denote the minimum degree sum of a pair of nonadjacent vertices. We conjecture that if |V(G)| = n = Σ^k_i=1 a_i and σ₂(G) ≥ n+k - 1, then for any k vertices v₁, v₂, . . . , v_k in G, there exist vertex-disjoint paths P₁, P₂, . . ., P_k such that | V(P_i)| = a_i and v_i is an endvertex of P_i for 1 ≤i≤ k. In this paper, we verify the conjecture for the cases where almost all a_i≤5, and the cases where k≤3.