In a way that is believable, trustworthy, or convincing; with credibility.
From 'credible' (believable) with the adverbial suffix '-ly.' Standard English formation from Latin 'credibilis.'
The phrase 'credibly alleged' shows up constantly in news writing—it's journalist-speak for 'we have enough evidence that readers should believe this, but we're not certain it's true,' making it a legal and ethical safety valve!
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