Luo–Tian–Wu's bipartite connectivity-keeping tree conjecture

Throughout, graphs are finite, simple, and undirected. Let $\k$ be a positive integer, and let $\T$ be a tree with bipartition $(X,Y)$. A graph is $\k$-connected when its connectivity satisfies $\kappa(G)\geq k$; $\delta(G)$ denotes minimum degree, and $\G-V(T')$ is the graph obtained by deleting the vertices of $\T'$.

Luo–Tian–Wu's conjecture. Every $\k$-connected bipartite graph $\G$ with

$$\delta(G)\geq k+\max\{|X|,|Y|\}$$

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

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

This is the bipartite analogue of Mader's connectivity-keeping tree conjecture. The conjecture remains open for $\k\geq 4$; the corresponding path result is known under a related minimum-degree condition.