Hasunuma's conjecture on removable trees in k-connected graphs

All graphs are finite, undirected, and simple. Let $k,n 1$, let $T$ be a tree of order $n$, and let $G$ be a $k$-connected graph. An edge set is deleted from $G$ by removing all its edges; in particular, $G-E(T')$ denotes the graph obtained by deleting the edges of a copy $T'$ of $T$.

Hasunuma's conjecture. If

$$\delta(G)\ge k+n-1,$$

then $G$ contains a copy $T'$ of $T$ such that $G-E(T')$ remains $k$-connected.

Hasunuma proved this for $k\le 2$ and when $T$ is a path; the case $k=3$ was proved independently by Liu, Liu, and Hong and by Yang and Tian. The general conjecture remains open.