The disk-potential deformation conjecture for knot augmentations

Let $K$ be a knot. For every Reeb chord $\alpha_{ij}$ connecting points $\xi_i$ and $\eta_j$, let $\alpha_{ij}(t,p)$ be a function, and write $\epsilon=\{\alpha_{ij}(t,p)\}$ for the resulting collection. Let $U_{K}^{0,\epsilon}(t,p)$ denote the disk potential obtained by counting generalized disks with boundary on $M_K$ and negative punctures at these Reeb chords, weighting each negative puncture asymptotic to $\alpha_{ij}$ by $\alpha_{ij}(t,p)$. Set

$$x=\frac{\partial U_{K}^{0,\epsilon}}{\partial p}.$$

Disk-potential deformation conjecture. For any knot $K$, there is a collection of functions $\epsilon=\{\alpha_{ij}(t,p)\}$ such that $U_{K}^{0,\epsilon}(p,t)$ is invariant under deformations, and the equation defined by $x$ cuts out a branch of the augmentation variety. Moreover,

$$\lim_{t\to 0}\exp\left(\partial_tU_{K}^{0,\epsilon}(p,t)\right)=(1-e^p)^{-1}\Delta_K(e^p).$$

This conjecture proposes that the Alexander polynomial of an arbitrary knot is recovered from a deformation of the disk potential incorporating negative punctures at Reeb chords. The invariance and augmentation-variety statement, as well as the required deformation and its limiting behavior, are not established in the supplied text.