In [Graphs Combin.~24 (2008) 469--483.], the third author and the fifth author conjectured that if $G$ is a $k$-connected graph such that $σ_{k+1}(G) \ge |V(G)|+κ(G)+(k-2)(α(G)-1)$, then $G$ contains a Hamiltonian cycle, where $σ_{k+1}(G)$, $κ(G)$ and $α(G)$ are the minimum degree sum of $k+1$ independent vertices, the connectivity and the independence number of $G$, respectively. In this paper, we settle this conjecture. This is an improvement of the result obtained by Li: If $G$ is a $k$-connected graph such that $σ_{k+1}(G) \ge |V(G)|+(k-1)(α(G)-1)$, then $G$ is Hamiltonian. The degree sum condition is best possible.