The evaluation-map conjecture for the HOMFLYPT difference module

Let $L$ be a link, let $\mathsf{H}_q(L)$ be its HOMFLYPT difference module, let $[L]$ be its distinguished link class, and let $\mathsf{ev}\colon \mathsf{H}(L)\to \mathcal{F}_r$ be the evaluation map. Let $I_q(L)$ be the annihilator of $[L]$ and let $I_q'(L)$ be the annihilator of $\mathsf{ev}([L])$. Evaluation-map conjecture. The map $\mathsf{ev}$ is injective and

$$I_q(L)=I_q'(L).$$

This is the formal conjecture underlying the claim that the HOMFLYPT difference module captures exactly the relations among antisymmetric colored HOMFLYPT polynomials; its resolution would also clarify the relation between the quantum recursion ideal and its classical augmentation ideal. The supplied text gives no resolution.