A mentorship program might count toward contributions to the profession, but it won't count toward professional experience. To count as professional experience, the work would have to be paid. Unpaid internships don't count, paid internships do count as professional experience.
The only website you should be referring to about this is the GISCI website. AAG and ESRI don't determine what can contribute to your GISP portfolio, only the GISCI can make that determination.