Nah, he's cool with them
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Umm, I don't know where you got that she thought he was a spy (point me in the right direction if I'm missing something). He's one of the people she strongly suspects as the murderer/in cahoots with the murderer, I guess, but otherwise it's mostly "DO YOUR JOB FURNITURE >:O."
Actually it's kind of implied that Natsuhi sometimes wants the servants to spy on Jessica and tell Natsuhi when she's acting inappropriately XD.