At any time more than 20 days before any contested hearing held pursuant to HRS sections 571-11 to 14 (excluding law violations, criminal matters, and child protection matters) is scheduled to begin, any party may serve upon the adverse party an offer to allow a judgment to be entered to the effect specified in the offer. Such offer may be made as to all or some of the issues, such as custody and visitation. Such offer shall not be filed with the court, unless it is accepted. If within 10 days after service of the offer the adverse party serves written notice that the offer is accepted, any party may then file the offer and notice of acceptance together with proof of service thereof and thereupon the court shall treat those issues as uncontested. An offer not accepted shall be deemed withdrawn and evidence thereof is not admissible, except in a proceeding to determine costs and attorney’s fees. If the judgment in its entirety finally obtained by the offeree is patently not more favorable than the offer, the offeree must pay the costs, including reasonable attorney’s fees incurred after the making of the offer, unless the court shall specifically determine that such would be inequitable.