A monotonicity conjecture for the local maximal singularity of the Hilbert scheme of points

The Brian\c{c}on-Iarrobino conjecture predicts the maximum singularity of the Hilbert scheme of a tetrahedral number of points. As for the maximal singularities of the Hilbert scheme of a non-tetrahedral number of points, the second named author gave some separate conjectural necessary and sufficient conditions. In this paper, we provide a conjectural sufficient condition for the necessary condition, and propose a monotonicity conjecture which predicts that for a fixed colength $l$, the maximal dimension of the tangent space over all the Borel-fixed ideals of colength $l$ is increasing with respect to the smallest pure exponent of the ideal.