Valid mining claims within the watershed of the headwaters of the Bonito River in the Lincoln National Forest, within the State of New Mexico, as above described, existing on June 13, 1939, and thereafter maintained in compliance with the law under which they were initiated and the laws of the State of New Mexico, may be perfected under sections 482e and 482f of this title, or under the laws under which they were initiated, as the claimant may desire.