The identification of the first Bar-Natan–van der Veen polynomial with the reduced 2-loop polynomial

Let $K$ be a 0-framed knot, and let $\rho_K^{1,0}$ be its first Bar-Natan–van der Veen polynomial. Let $\Theta_K(t_1,t_2)$ be the 2-loop polynomial, and define

$$\widehat{\Theta}_K(t):=\Theta_K(t,1)\in\mathbb{Q}[t,t^{-1}].$$

Reduced 2-loop identification conjecture.

$$\rho_K^{1,0}=\widehat{\Theta}_K.$$

The source states that the $\mathfrak{sl}_2$ weight system gives $\widehat{\Theta}_K=P_K^1$, so this is equivalent to the first-polynomial identification conjecture above. It is therefore recorded as a restatement rather than a separate mathematical claim.