Shi–Shan's toughness conjecture for forbidden linear forests

Let $k\ge 4$ be an integer. A graph is $t$-tough if its toughness satisfies $\tau(G)\ge t$, a graph is $r$-connected if deleting fewer than $r$ vertices leaves it connected, and $P_n$ denotes the path on $n$ vertices; $kP_1$ is the disjoint union of $k$ isolated vertices. A graph is $(P_2\cup kP_1)$-free if it has no induced subgraph consisting of a two-vertex path together with $k$ isolated vertices. Shi–Shan's conjecture. Let $G$ be a $1$-tough and $2k$-connected $(P_2\cup kP_1)$-free graph. Then $G$ is hamiltonian. This conjecture is a strengthening of the general toughness conjecture for Hamiltonicity in a structured graph class. It has been independently confirmed by Xu, Li, and Zhou, and by Ota and Sanka.