If it was introduced by the Mamluks, sure, but not until 1200AD or so
Perhaps through the audaciousness of Qutuz asking for Christian help in his battle of the Mongols there was a trading of...gambling games? A long shot...
I think it started as a game: either a child's or gambling. I am not sure if the European version came from the Mongols via China (eventually) or from Islam via china, Mongols and then Mamluks, or even some other order of the above route/choices. I am convinced that with a few exceptions it reached Europe as a deck they would understand and was converted as such either in Qaraqorum ( a personal little indulgence of mine) or along one of the many Islamic or Mongol borders with Europe/ Eastern Christianity in the 12th century. Europe was a cultural backwater, selfish, internal fighting machine during the 11, 12 and into the 13th century,offering little to the world to trade or conquer. The Pope and the holy Roman Empire could not see eye to eye and all kingdoms in Europe were influenced by this ongoing political/religious struggles in Europe as far as I am concerned. Anything that made it back to us was though traders, monks or crusaders. I think we can take Monks off the list
I am certain it came out of Persia translated and redrawn to a recognisiabe iconic/imagery deck of cards by the time it hit Europe. Sure a few curiosity decks made it in the form of "mamluk" cards and variations, but there is no reason for any of these few curiosities to survive in Europe.