The pension board shall determine the eligibility of each employee for membership in the system based upon information furnished by Jefferson County or its designated agent. Such determination shall be conclusive and binding upon all persons. Additionally, if the pension board makes a determination upon an employee's date of hire by the employer that such employee is an eligible employee, then such employee shall continue to participate in the plan, even if such employee subsequently no longer meets the definition of an eligible employee under Section 45-37-123.01, unless the pension board's initial determination was erroneous.