Alexander-polynomial formula for the proper Abelian BF knot observable

Let $K$ be a knot, let $\lambda$ be the coupling parameter, and let $\bigcirc$ denote the unknot. Let $\Delta(K;z)$ be the Alexander polynomial evaluated at $z$, and set

$$z(\lambda)=2i\sin(\lambda/2).$$

Let ${\cal O}[K,\lambda]$ be the proper solution whose vacuum expectation value is ambient-isotopy invariant. Alexander-polynomial conjecture. Its normalized vacuum expectation value satisfies

$$\frac{\left\langle{\cal O}[K,\lambda]\right\rangle_\lambda}{\left\langle{\cal O}[\bigcirc,\lambda]\right\rangle_\lambda}=\frac{1}{\Delta(K;z(\lambda))}.$$

This identifies the normalized Abelian BF observable with the inverse Alexander polynomial, extending the classical observable's expected knot-invariant formula to the quantum setting.