When a petition is filed as authorized in Arkansas Constitution, Amendment 38, as amended by Amendment 72, to submit to the electors of a county the question of levying, increasing, decreasing, or repealing a county library tax and the petition or the sponsor of the petition requests that the question be submitted at a special election, the sponsor of the petition may be required by the county quorum court to pay a filing fee not to exceed two thousand dollars ($2,000) which shall be used to offset a portion of the cost of calling and conducting the special election.