The $\frac{5}{4}$-tough spanning 2-trail conjecture for $2K_2$-free graphs

A graph is $2K_2$-free if it has no induced subgraph consisting of two independent edges, and it is $t$-tough if $|S|\geq t\,c(G-S)$ whenever deletion of $S$ leaves at least two components. A spanning 2-trail is a spanning trail with maximum degree at most $4$.

The $\frac{5}{4}$-tough spanning 2-trail conjecture. Any $\frac{5}{4}$-tough $2K_2$-free graph with at least three vertices has a spanning 2-trail.

The paper constructs $2K_2$-free graphs with toughness approaching $\frac{5}{4}$ that have no spanning 2-trail, motivating this threshold conjecture; no proof or refutation is given in the supplied text.