Connectivity keeping edges of trees in 3-connected or 3-edge-connected graphs

Hasunuma [J. Graph Theory 102 (2023) 423-435] conjectured that for any tree $T$ of order $m$, every $k$-connected (or $k$-edge-connected) graph $G$ with minimum degree at least $k+m-1$ contains a tree $T^{\mathrm{\prime }}\cong T$ such that $G-E(T^{\mathrm{\prime }})$ is still $k$-connected (or $k$-edge connected). Hasunuma verified this conjecture for $k\le 2$. In this paper, we confirm this conjecture for $k=3$.