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The "begging the question" issue, I believe, is pretty much over.
Educated people, or people who are just interested in accuracy, know that the expression "begging the question" has, for centuries, designated the dishonest process of framing a statement that pretends to be a question but is actually some kind of device intended to evoke or force a desired answer.
More recently, uneducated people have heard the expression and, incorrectly but understandably, misinterpreted it as designating a situation that suggests that a particular question be addressed. The expression, in its new but incorrect usage, has become quite common. I heard it twice today.
I agree that the new interpretation seems viscerally to be more correct than the prior one. But logicians need to come up with a replacement for the hijacked term as quickly as possible. I would hate for a specific tool for duplicity to go for very long without its own designation and its own recognition.
Update: I recently heard someone say "...prompts the question: ..."