Any licensee who fails to renew his or her license by June 30 of the year in which the licensee’s renewal is set to occur shall be required to pay double the renewal fee if paid within 90 days of the expiration date. Any failure, neglect, or refusal on the part of any person licensed by the board to renew the license as provided in N.H. Rev. Stat. § 329:16-a or this section shall automatically lapse such license. Licenses lapsed under this section for nonpayment within 90 days shall not be reinstated except upon payment of a reinstatement fee as established by the board, and a showing of such evidence of professional competence as the board may reasonably require.