The Secretary of the Interior (a) shall, whenever it shall be shown to his satisfaction that a tract of public land has been held in good faith and in peaceful, adverse, possession by a claimant, his ancestors or grantors, under claim or color of title for more than twenty years, and that valuable improvements have been placed on such land or some part thereof has been reduced to cultivation, or (b) may, in his discretion, whenever it shall be shown to his satisfaction that a tract of public land has been held in good faith and in peaceful, adverse, possession by a claimant, his ancestors or grantors, under claim or color of title for the period commencing not later than January 1, 1901, to the date of application during which time they have paid taxes levied on the land by State and local governmental units, issue a patent for not to exceed one hundred and sixty acres of such land upon the payment of not less than $1.25 per acre: Provided, That where the area so held is in excess of one hundred and sixty acres the Secretary may determine what particular subdivisions, not exceeding one hundred and sixty acres, may be patented hereunder: Provided further, That coal and all other minerals contained therein are reserved to the United States; that said coal and other minerals shall be subject to sale or disposal by the United States under applicable leasing and mineral land laws, and permittees, lessees, or grantees of the United States shall have the right to enter upon said lands for the purpose of prospecting for and mining such deposits: And provided further, That no patent shall issue under the provisions of this chapter for any tract to which there is a conflicting claim adverse to that of the applicant, unless and until such claim shall have been finally adjudicated in favor of such applicant.