The perturbed Conway invariant positivity conjecture

For a knot $K\subset S^3$, let $\rho_1(K)$ be an invariant divisible by $(1-T)^2$ and symmetric under $T\mapsto T^{-1}$. Define

$$\rho_1^{\mathrm{red}}(K)=\frac{T}{(1-T)^2}\rho_1(K)$$

and let the perturbed Conway invariant $\delta_1(K)\in\mathbb{Z}[z]$ be the unique polynomial satisfying

$$\delta_1(K)(x-x^{-1})=\rho_1^{\mathrm{red}}(K)(x^2).$$

Perturbed Conway invariant positivity conjecture. If $K\subset S^3$ is a positive knot, then all coefficients of $\delta_1(K)$ are non-positive.

This is motivated by Cromwell's theorem that the coefficients of the Conway polynomial of a positive knot are non-negative. The divisibility and symmetry assumptions on $\rho_1$ have been conjectured by Bar-Natan and van der Veen and verified for knots with up to 11 crossings; the sign assertion for $\delta_1$ is presented here as a conjecture.