Shi and Shan's conjecture for tough connected $(P_2\cup kP_1)$-free graphs

Let $k$ be an integer with $k\ge 4$, and let $G$ be a graph. Here $P_n$ denotes the path on $n$ vertices, $P_2\cup kP_1$ is the disjoint union of a two-vertex path and $k$ isolated vertices, a graph is $1$-tough when its toughness is at least $1$, and $2k$-connected when deleting fewer than $2k$ vertices leaves it connected. The graph is $(P_2\cup kP_1)$-free if it has no induced subgraph isomorphic to $P_2\cup kP_1$. Shi and Shan's conjecture. Every $1$-tough, $2k$-connected $(P_2\cup kP_1)$-free graph is Hamiltonian. Shi and Shan had proved the analogous statement with $4$-toughness for $k\ge4$; the conjecture weakens the toughness requirement to $1$ and remains open in the supplied source.