Connectivity keeping pendant extensions of paths in $k$-connected triangle-free graphs

Motivated by Mader's conjecture on connectivity keeping trees, we study the corresponding problem in triangle-free graphs. For positive integers $k,m$, we first prove that every $k$-connected triangle-free graph $G$ with $δ(G)\geq k+\left \lceil \frac{m}{2} \right \rceil $ contains a path $P$ of o