The rank-matrix characterization conjecture for codimension-three Artinian Gorenstein algebras

Let $M$) be an upper triangular matrix of size $d+1$ with non-negative entries. For an upper triangular matrix, let $\operatorname{diag}(i,M)$ denote its $i$-th diagonal vector, and let $\left(\operatorname{diag}(i+1,M)\right)_+$ denote the corresponding positive-part vector. Let $A$ be an Artinian Gorenstein algebra of codimension three with Hilbert function $h_A$, and let $\ell\in A_1$ be a linear form. A sequence is an O-sequence when it is the Hilbert function of a standard graded algebra. Rank-matrix characterization conjecture. The matrix $M$ is the rank matrix of some such pair $(A,\ell)$ if and only if: (i) for every $0\leq i\leq d$, $\operatorname{diag}(i,M)$ is an O-sequence and $h_A=\operatorname{diag}(0,M)$; (ii) for every $0\leq i\leq d-1$, the difference vector $\operatorname{diag}(i,M)-\left(\operatorname{diag}(i+1,M)\right)_+$ is an O-sequence; and (iii) every $2\times 2$ square submatrix of successive entries on and above the diagonal, of the form

$$\begin{pmatrix}u&v\w&z\end{pmatrix},$$

satisfies $w+v\geq u+z$. Based on computations for many examples through socle degree nine, the necessary conditions in the conjecture have not been shown to fail; the claim proposes that they are also sufficient for realizing the rank matrix.