The diagonal determination conjecture for Bar-Natan–van der Veen polynomials

Let $K$ be a 0-framed knot, and let $\rho_K^{i,i}$, $i>0$, denote the diagonal family of Bar-Natan–van der Veen polynomials. Let $\Delta_K$ be the Alexander polynomial of $K$.

Diagonal determination conjecture. The family of polynomials $\rho_K^{i,i}$, $i>0$, is completely determined by the Alexander polynomial of $K$.

The cases $\rho_K^{1,1}$ and $\rho_K^{2,2}$ are shown in the paper to depend only on the Alexander polynomial; the conjecture proposes the same for every diagonal term.