Chvátal's spanning closed 1-trail conjecture

Let a $t$-tough graph $G$ satisfy $\omega(G\setminus S)\leq\max\\{1,\frac{1}{t}|S|\\}$ for every $S\subseteq V(G)$. A spanning closed $1$-trail is a spanning closed trail meeting each vertex at most once, hence a Hamiltonian cycle. Chvátal's conjecture. There exists a positive real number $t_0$ such that every $t_0$-tough graph of order at least three admits a spanning closed $1$-trail. This is the explicitly displayed reformulation of the Hamiltonicity conjecture in the source; the source discusses only a weaker confirmed result.