A. When the thing held in indivision is not susceptible to partition in kind, the court shall decree a partition by licitation or, as provided in Paragraph B of this Article, by private sale and the proceeds shall be distributed to the co-owners in proportion to their shares.
B. In the event that one or more of the co-owners are absentees or have not consented to a partition by private sale, the court shall order a partition by private sale and shall give first priority to the private sale between the existing co-owners, over the sale by partition by licitation or private sale to third persons. The court shall order the partition by private sale between the existing co-owners as identified in the conveyance records as of the date of filing for the petition for partition by private sale. The petition for partition by private sale shall be granted first priority, and the sale shall be executed under Title IX of Book VII of the Code of Civil Procedure.