Here's how to demonstrate this.
Make the CQC leagues a more prominent part of the game/setting.
I don't just mean things like "support your faction's happiness by doing CQC matches while parked at their stations".
I'm talking an actual CQC league in-game that's hosted in a different system each week (possibly determined by which faction the winners of the previous week support?) and spawns CZ-like instances. Call them "CQC Arena [no holds barred]" or something. The instant you pledge to a team, the zone becomes lawless. You get paid bonds for blowing stuff up. Maybe scenarios that trigger once there are players on both teams. Allow people to spawn into the actual in-game arena instances from the CQC menu. Whatever other nonsense you want to add, but the long and the short of it would be "here are lawless zones where you get paid for going in and blowing up other player-controlled ships, and an in-game announcement every week to tell the PvPers what system to head towards".
I honestly wonder how many would actually show up.
Depends, if it was available in PGs, then I know who wouldn't be seen in open doing it
But I like the idea of repurposing CQC to a main game, open only, CZ style of play.
I think that could be fun, if teams were balanced properly.
In fact, those who joined from the CQC menu could be placed to make up the numbers in the smaller team and those who do it from the system gets to choose.
That way, those who care about the faction, go and support it and those who just want a scrap can use the menu and end up anywhere.