Combinatorial criteria for maximal tangent spaces of Borel-fixed ideals

Let $N\geq 3$. Let $I$ be a zero-dimensional Borel-fixed ideal in $\mathbb{C}[x_1,\ldots,x_N]$ whose convex hull is spanned by its monomial generators. Suppose that

$$I=(x_1^{m_1},x_2^{m_2},\ldots,x_N^{m_N},\text{all the mixed monomial generators}),$$

where $m_1\leq\cdots\leq m_N$ and, if $\binom{N-1+k}{N}\leq\operatorname{colength}(I)<\binom{N+k}{N}$, then $m_1=k$. Assume that one of the three detailed alternatives (I), (II), or (III) in the source holds, concerning the multiplicities $m_i$, the lower and upper boundaries of $\operatorname{conv}(I)$, their lattice points and symmetry, and the location of the monomials of $I$. The combinatorial criteria conjecture. For $N=3$, the ideal $I$ has the maximum dimension tangent space among all elements of $\operatorname{Hilb}^{\operatorname{colength}(I)}(\mathbb{A}^N)$. The conjecture is intended to characterize the most singular points of Hilbert schemes of points; the source further asks whether it holds for arbitrary $N\geq 3$.