If the cardboard is safe for unexposed paper it certainly is safe for a finished print.
Unexposed paper is not intended to be stored for many years, and materials for the boxes are selected accordingly to serve the short-term packaging function but otherwise to minimize cost.
Sometimes one can get away with using the paper boxes, but it's impossible to know for sure up front. So it's a question of how highly one values one's work, how long one intends for it to last and how much one is willing to invest in minimizing known risks to long-term stability. We all have limits to what we can afford and how much we're willing to fuss.
RC papers of today are different from the original RC papers. Also, the deterioration mechanisms that are most important for a framed RC print on display are not identical to those that are most prominent in dark storage.
FWIW, I store most of my RC prints, all selenium toned, loose in acid-free storage boxes. If they were going to be handled frequently, I might include acid-free tissue interleaving. It's not expensive at all per print, taking into account the materials cost and amount of my effort that goes into them. I use polypropylene sleeves extensively for negatives, occasionally for prints under special circumstances.