Most probably going to be a physical issue with the adaptor. It could be as simple as needing to give it a little more force to plug it in just that little bit further and make better contact, or it could be something more awkward like misaligned contacts or a ground loop.
You mention that she already has a similar headset which produces the same basic issue. Does it include a similar adapter, or have your tests all been conducted with the one splitter? Have you also tried the same splitter on a different system?
Give it a try only hooking up the microphone portion of the splitter - leave the headphone jack just hanging in the air (but not touching on anything metal). Use a different headset if you still require sound output, but since you’re broadcasting you can listen to the result on a different machine. If that works clearly than odds are it’s a ground loop, which fortunately can be solved (though cost may or may not be worth it).