About vertices of degree $k$ of minimally and contraction critically $k$-connected graphs: upper bounds

In the article [4], R. Halin asked, what the constant $c_k$ such that any minimally and contraction critically $k$-connected graph has at least $c_k|V(G)|$ vertices of degree $k$. Exact bound for $k=4$ ($c_4=1$) and no upper bound for larger $k$ is known now. We found upper bounds for $c_k$ for $k\geq5$.