The Partner/Conspirator badge is for completing Weekly Strike Targets when you've already earned the reward for the week, 50 times to be precise. This week the blueside WST is the Cavern of Transcendence trial. This trial is easily soloed since the removal of simultaneous glowie clicks from the game. Any two of Stealth/Super Speed/Stealth IO will allow any character to sneak past the enemies to grab the glowies, and a tray full of Lucks can let you grab the glowies if a mob happens to spawn near them. If you don't have stealth capabilities on your badger, you can always switch to a second or third build and pick them up there. Once all eight glowies are collected, all you have to do is run past the boss enemy in the last room and defeat the mobs guarding the hostage in back. The whole trial can be done in 5-10 minutes, and you can cut down the time on (some) runs by using the various mission door teleporter powers (though you can easily finish before they recharge). After you finish the trial, you can solo queue in the LFG menu for the trial to get back to Karsis and restart immediately. This is a great opportunity to grind out WSTs to get that badge.
You do need to run the story arc for Talshak the Mystic to unlock Karsis for the trial, but since that's part of another badge I assume the for badging community this post is targeted towards that won't be an issue.