Mader's connectivity-keeping tree conjecture

Throughout, graphs are finite, simple, and undirected; $\k$ and $\m$ are positive integers, the order of a graph is its number of vertices, and $\kappa(G)$ denotes its connectivity. A graph is $\k$-connected when $\kappa(G)\geq k$. A tree has order $\m$ when it has $\m$ vertices.

Mader's conjecture. For any tree $\T$ of order $\m$, every $\k$-connected graph $\G$ with minimum degree

$$\delta(G)\geq \left\lfloor\frac{3k}{2}\right\rfloor+m-1$$

contains a subtree $\T'\cong T$ such that

$$\kappa(G-V(T'))\geq k.$$

Mader proved the corresponding assertion for a path, and later obtained a weaker quadratic minimum-degree bound for arbitrary trees. The conjecture is known for $\k\leq 3$ and remains open for $\k\geq 4$; a linear minimum-degree sufficient condition is known.