Embedded minimal disks with prescribed curvature blowup

We construct a sequence of compact embedded minimal disks in a ball in $\mathbb{R} ^3$, whose boundaries lie in the boundary of the ball, such that the curvature blows up only at a prescribed discrete (and hence, finite) set of points on the $x_3$-axis. This extends a result of Colding and Minicozzi, who constructed a sequence for which the curvature blows up only at the center of the ball, and is a partial affirmative answer to the larger question of the existence of a sequence for which the curvature blows up precisely on a prescribed closed set on the $x_3$-axis.