You know wrong.
From the National Catholic Register:
As early as Egeria’s day (circa 381), a relic of the True Cross was coveted by Christians, which would lead in years to come to a host of false relics, some the product of outright fraud, others of wishful thinking. Today, it is virtually impossible to distinguish which relics of the cross are genuine, although the relics displayed in the Church of the Holy Sepulcher in Jerusalem and Rome’s Basilica of the “Holy Cross in Jerusalem” are probably authentic.