Polynomial codimension conjecture for Jordan net orbits

Let $A^{(2)}_{r,k_1,k_2}$, $A^{(3)}_{k_1,k_2,k_3}$, and $B^{(2)}_{k,\ell_1,\ell_2}$ be the indicated Jordan nets in $\operatorname{Gr}(3,\operatorname{Sym}^n)$. Their codimensions are functions of their parameters.

Polynomial codimension conjecture. The codimensions of $A^{(2)}_{r,k_1,k_2}$, $A^{(3)}_{k_1,k_2,k_3}$, and $B^{(2)}_{k,\ell_1,\ell_2}$ in $\operatorname{Gr}(3,\operatorname{Sym}^n)$ are polynomial functions $f_2(r,k_1,k_2)$, $f_3(k_1,k_2,k_3)$, and $g(k,\ell_1,\ell_2)$, respectively, each of degree at most $2$.

This conjecture predicts uniform polynomial formulas for the codimensions of the remaining orbit families, extending the explicitly computed examples. The source provides no resolution, so the conjecture remains open.