Mou and Pasechnik's Hamiltonicity conjecture for tough $2K_2$-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 graph is hamiltonian if it has a spanning cycle.

Mou and Pasechnik's conjecture. Every $2$-tough $2K_2$-free graph with at least three vertices is hamiltonian.

The source presents this as one of two conjectures proposed by Mou and Pasechnik and gives no resolution for it here.