Partial classification conjecture for maximal tangent spaces in three variables

Let $k\geq 1$ and let $n=\binom{k+2}{3}+i$, where $i\in\{0,1,2,3,4,k+1,2k+1,\binom{k+3}{3}-\binom{k+2}{3}-1\}$ and $n<\binom{k+3}{3}$. Partial-classification conjecture. For each such $n$, there exists an ideal satisfying the conditions of the combinatorial criteria conjecture whose tangent space has maximum dimension among all elements of $\operatorname{Hilb}^n(\mathbb{A}^3)$. The claim records conjectural maximal examples for the listed colengths; the source does not establish it in general.