Isn’t sharing it publicly already a counter argument to “I don’t care if nobody knows about it”? Thus you wouldn’t have to falsify.

We could think of a scenario where his neighbor shares information about his (the authors) garden. In this case, if the author would seek measures to delete these posts, it might be interpreted as “signaling that he does not care and even doesn’t want others to know”.

I agree with you that signaling always shows intention (ie wanting something). You can “want not to want something”, though (: