"Claim" is my favorite of these words.

Main example: "Elected official claims nothing bad was done"

It has a gentle enough negative connotation to allow you to massage the emotions of your reader, while still being a technically accurate statement.

Only people with something to hide will "claim" or "deny" something. It's the same idea where you're talking about something and your audience can feel a "big old but" coming. Even if you never produce the but, they still have the feeling what was just stated is about to be refuted.