I've had some ideas in the past about a GJ faction. It's the kind of thing that could be unique and fun if the leadership is there to make it happen.
Basically my view of the GJ is kind of a "rough around the edges new rich" type. Kinda like the stereotypical cowboy millionaire in the western US. These guys are a little rough, a little crude, but still exceptionally wealthy and, until recently, favored by the Crown. I had kind of envisioned their leadership as being these really shadowy types used to pulling strings throughout Gallia.
Basically the GJ can do all sorts of things. They can spy in Sirius. They can aim to regain their close ties to the Crown. They can focus on just making money. They could smuggle.
Best of luck to anyone who wants to go for it with them. I'd try it myself if I had some more time.