Liens under AS 34.35.125, 34.35.130, and 34.35.140 are security for payment for the work performed within a period of nine months immediately before the cessation of the work. The liens bind all the right, title, and interest in the property mentioned of the person at whose instance or request or for whom the work is performed, to the full extent of the interest that the person has at the beginning of the work for which the lien is claimed, or subsequently acquired in the property, up to the time of the foreclosure of the lien.