The classical-limit conjecture for the augmentation ideal

Let $L$ be a link, let $I_q(L)$ be the ideal of quantum relations annihilating the distinguished class $[L]$, let $I_q'(L)$ be the annihilator of its evaluation in antisymmetric colored HOMFLYPT polynomials, and let $I(L)$ be the base change of $I_q(L)$ to $q=1$. Let $\mathsf{Au}_L$ be the augmentation ideal of $L$, whose vanishing set lies in $(\mathbb{C}^{\times})^{2r}$. Augmentation-ideal conjecture. For every link $L$,

$$I_q(L)=I_q'(L),\qquad I(L)=\mathsf{Au}_L.$$

Equivalently, $\mathsf{Au}_L$ is the classical limit of the ideal defining recursion relations for antisymmetric HOMFLYPT polynomials; in particular, its vanishing set is Lagrangian in $(\mathbb{C}^{\times})^{2r}$. The text explains that this would follow from the vanishing of a connecting homomorphism together with the evaluation conjecture, but does not establish it.