The refined square-zero upper-triangular matrix conjecture

Let $k$ be an algebraically closed field, let $r$ be a positive integer, let $N=2n$ be an even positive integer, and let $d=(d_1,\ldots,d_N)$ be an $N$-tuple of nonincreasing integers. Let $V(d)$ be the projective variety of square-zero strictly upper-triangular $N\times N$ matrices with weights $\deg(x_{ij})=d_i-d_j+1$, and let $L(d)$ be the subvariety of matrices of rank less than $n$. For $1\leq\mathcal{R},\mathcal{C}\leq N$, let $V(d)_{\mathcal{R}\mathcal{C}}$ be the subvariety defined by $x_{ij}=0$ whenever $i\geq N-\mathcal{R}+1$ or $j\leq\mathcal{C}$. Refined square-zero matrix conjecture. If there exists a nonconstant morphism $\psi:\mathbb{P}^{r-1}_k\to V(d)_{\mathcal{R}\mathcal{C}}-L(d)$, then $N\geq 2^{r-1}(\mathcal{R}+\mathcal{C})$. This conjecture is stated as a strengthening of the square-zero matrix conjecture; the paper proves the corresponding results for the small cases described in its abstract, while the general assertion remains open.