@Cogsworth THANK YOU!!! This fixed what my can of WD-40 Contact Cleaner couldn’t. I lucked out and found a set at my local Goodwill a few days ago, but they had this problem along with some serious crackle. The cleaner helped the crackle problem, and the re-solder helped the rest.
For those with the round PS/2-style connector for your control pod, this is likely your only option outside 3rd-party replacements, as Klipsch support told me only the newer, VGA-style connector is available on the replacement part.