Within 20 years from and after the adoption of any plat by Council, no permit shall be issued by the Director of Buildings for the erection of any building, structure, or part thereof on any part of the land between the proposed lines of a street or streets, as platted upon any approved plat, and no building, structure, or part thereof shall be erected on any such land. However, the Zoning Board of Appeals shall have the power, upon an appeal filed with it by the owner of any such land or part thereof, by a vote of a majority of its members, to direct the Director of Buildings to grant a permit for a building, structure, or part thereof within the platted street location. The Board shall make this direction in any case in which it finds, upon the evidence and arguments presented to it upon the appeal, that, balancing the interest of the municipality in preserving the integrity of the municipality’s plan, as evidenced by the plat, against the interest of the owner of the property in the use of his or her property and in the benefits of the ownership thereof, the grant of the permit is required by considerations of reasonable justice and equity.