In the multiterminal cut the input is a graph $G$ and a subset $T$ of its vertices. The task is to remove the minimum number of edges from $G$ such that there is no path connecting any distinct vertices of $T$. The problem is APX-hard, even when $T$ has 3 vertices. This was shown in by Dalhaus et al (E. Dahlhaus, D. S. Johnson, C. H. Papadimitriou, P. D. Seymour, M. Yannakakis: The Complexity of Multiterminal Cuts. SIAM J. Comput. 23(4): 864-894, 1994.)
I am interested in the actual value $\alpha>1$ such that no $(\alpha-\varepsilon)$-approximation is possible for multiterminal cut, unless P=NP. A quick adaptation of the reduction used by Dalhaus et al. for general $T$ implies $\alpha=89/88$. (For $|T|=3$ the constant is worse, but I am interested in arbitrary $|T|$.)
My question is: has there been any improvement over Dalhaus et al for the inapproximability of multiterminal cut? Are there other NP-hardness proofs of multiterminal cut in the literature?
Since there are constant-factor approximations for the problem, the question here is about constants.