Fujita's connectivity-preserving connected subgraph conjecture

Let $k$ and $m$ be positive integers. A graph $G$ is $k$-connected when $ba(G)\geq k$, where $ba(G)$ is the minimum size of a vertex set whose deletion disconnects $G$ or leaves only one vertex; write $b4(G)$ for the minimum degree, and let $|W|$ denote the order of a graph or subgraph $W$. Fujita's conjecture. There is a least non-negative integer $f_k(m)$ such that every $k$-connected graph $G$ satisfying

$$b4(G)\geq \left\lfloor\frac{3}{2}k\right\rfloor-1+f_k(m)$$

contains a connected subgraph $W$ of exact order $m$ such that $G-V(W)$ is still $k$-connected. Fujita and Kawarabayashi proposed this as a degree threshold for connectivity-preserving connected subgraphs; it was later confirmed by Mader, who proved $f_k(m)=m$ and that $W$ can be chosen to be a path.