Table-locus complete-intersection conjecture

Fix $Q=(u,u-r)$ with $u>r\ge 2$, put $B=J_Q$, and let $\mathcal U_B$ be the maximal nilpotent subalgebra of the centralizer of $B$. For a partition $P$, let $\mathfrak Z(P)\subseteq\mathcal U_B$ be the locus of matrices of Jordan type $P$, and let $\mathcal I(\mathfrak Z(P))$ be its vanishing ideal. Table-locus conjecture. For every table entry $P=P_{k,\ell}$, the ideal $\mathcal I(\mathfrak Z(P))$ is an irreducible complete intersection generated by $k+\ell-2$ specific $sl_2$-homogeneous polynomials, of which $\min\{k+\ell-2,r-2\}$ are linear and the remainder are quadrics. Along a fixed A row or B/C hook, each successive locus is obtained by adjoining one equation; each new quadric on the $k+\ell=s+r$ diagonal is a sum of $s$ determinants of $2\times2$ matrices of variables and has $sl_2$ weight $2(k+\ell-1)$. These are proposed equations based on computations; the supplied text gives no resolution.