Jet graphs of very well covered graphs are very well covered

Let $G$ be a very well covered graph, and for $s\in\mathbb{N}$ let $\mathcal{J}_s(G)$ denote its $s$-th jet graph. A graph is very well covered if all its minimal vertex covers have the same cardinality and this cardinality is half the number of vertices. Jet-graph conjecture. For every $s\in\mathbb{N}$, the graph $\mathcal{J}_s(G)$ is very well covered. The conjecture is motivated by the fact that jets of the complete bipartite graphs $K_{n,n}$ are very well covered, and computational evidence shows that the jets up to third order of Favaron's example of a very well covered graph have the same property; the general case remains open.