Cliques in $k$-connected graphs

The existance of $n+1$-cliques in $k$-connected graphs is studied. It is proved that in a $k$-connected graph $G$ such a clique exists provided $G$ satisfies the following conditions: (1) the vertices of any $n$-clique of $G$ lie in a $k$-separating set; (2) after removing certain pairs, each consisting of a vertex and an edge, the connectivity of the graph $G$ decreases by 2.