Historic Routes Preservation Act This bill allows any state, county, political subdivision or agency of a state, company, or other person asserting the public acceptance of an open-ended grant or dedication of land by the United States for a right-of-way allowing public use and passage for the construction of highways over public lands to file a claim if such claimant, before enactment of this bill (1) filed a notice of an intent to file suit with respect to a claim asserting the public acceptance of such a right-of-way; or (2) filed suit with respect to such a claim under the Quiet Title Act. The bill sets forth (1) a procedure for conclusively verifying, proving, and establishing the acceptance of such rights-of-way; and (2) provisions governing judicial review of a denial of such a claim.