I’m somewhat skeptical of emphatic statements about that long ago.
Plausibly correct.
Possibly not AS correct as assumed.
Actually the history of the use of the cross as a symbol is better known than a lot of stuff. And, we are discussing something that Roman historians noted also.
I am not arguing against your skepticism, but we are talking about a period when there is a written recorded history.
Google "Christian cross history" and see what comes up.