Any person who, within a 36-consecutive-month period, has been convicted of two misdemeanor offenses under Chapter 11 (§ 4.1-1100 et seq.) of Title 4.1, subsection B of former § 18.2-248.1:1, or § 18.2-250 shall be ineligible to purchase or transport a handgun. However, upon expiration of a period of five years from the date of the second conviction and provided the person has not been convicted of any such offense within that period, the ineligibility shall be removed.