HOMFLY-PT specialization conjecture from the augmentation polynomial

Let $K$ be a knot in $S^3$, and let $f(U)$ be the polynomial determined by the branch of the zero locus of $\operatorname{Aug}_K$ near $(\lambda,\mu,U)=(0,U,U)$ through

$$\mu=U+f(U)\lambda+O(\lambda^2).$$

Here $f(U)$ can be expressed using the coefficients of $\operatorname{Aug}_K$ at powers $\lambda^1$ and $\lambda^0$. Let $P_K(a,q)$ denote the HOMFLY-PT polynomial of $K$. HOMFLY-PT specialization conjecture.

$$\frac{f(U)}{U-1}=P_K(U^{-1/2},1).$$

This gives a proposed relation between the first-order behavior of the augmentation variety and a specialization of the HOMFLY-PT polynomial. It had been checked in the source for all knots whose augmentation polynomial was known.