The link which you have provided is related to k-vertex connected spanning subgraph problem. My problem is different than k-connected vertex connected spanning subgraph.
k-vertex connected minimal Steiner network problem is defined as :
For an undirected and weighted graph G = (V,E) and a terminal set S subset of V , the k-connected Steiner minimal network (SMN) problem requires to compute a minimum-weight subgraph of G in which all terminals are k-connected to each other.
If I did understood your request. Given a weighted graph G = (V,E) and a set of terminals T. The problem concerns the minimum cost subgraph such that every pair of terminals are connected by k openly (vertex) disjoint paths,
I dont know if you did get the following reference from Bundit Laekhanukit :