Fortunately, Philip J. Davis wrote a paper on this exact topic in 1985: https://www.gwern.net/docs/www/www.maa.org/9230a8f2614ddd74b52c556fe0364fbcc24e1a30.pdf

Fortunately, Philip J. Davis wrote a paper on this exact topic in 1985: https://www.gwern.net/docs/www/www.maa.org/9230a8f2614ddd74b52c556fe0364fbcc24e1a30.pdf https://www.gwern.net/docs/www/www.maa.org/9230a8f2614ddd74b... He listed at least 21 states between "conjecture" and "proof".