Connectivity keeping stars or double-stars in 2-connected graphs

In Mader (2010), Mader conjectured that for every positive integer k and every finite tree T with order m, every k-connected, finite graph G with δ(G)≥⌊32k⌋+m−1 contains a subtree T′ isomorphic to T such that G−V(T′) is k-connected. In the same paper, Mader proved that the conjecture is true when T is a path. Diwan and Tholiya (2009) verified the conjecture when k=1. In this paper, we will prove that Mader's conjecture is true when T is a star or double-star and k=2.