Whenever any person upon whom is imposed any duty connected with the calling, warning, or conducting of any town, ward, or district meeting, or with the canvassing of the lists of voters for use at these meetings, or with the counting of the votes cast at the meetings, or the declaring of the result of the votes, fails, neglects, or refuses to perform his or her duty within the time specified by law for its performance, the supreme court shall immediately, upon the petition of any citizen of the city or town where the failure, neglect, or refusal occurred, issue its writ of mandamus ordering the person failing, neglecting, or refusing to perform the duties in question within twenty-four (24) hours from the time of the issuance of the writ; and if the failure, neglect, or refusal continues beyond the twenty-four (24) hours specified in the writ, the court shall immediately appoint some suitable person to at once perform the duties, and the person refusing to obey the writ of mandamus shall, in addition to the penalties by law for the failure, neglect, or refusal to perform his or her duties, be liable to those further penalties for contempt of court that the court may impose for failure to obey the writ.