The Lagrangian-reduction conjecture for link augmentation varieties

Let $K$ be a non-split link, let $K_n$ be its $n$th component, and let $K-K_n$ be the remaining link. Let $V_K(n)$ and $V_{K_n}(1)$ be the corresponding augmentation varieties, and define reduction along $V_{K_n}(1)$ by projecting the intersection to the coordinates of $K-K_n$. Lagrangian-reduction conjecture. $V_{K-K_n}(n-1)$ is contained in this reduction; concretely,

$$V_{K-K_n}(n-1)\subset\{(\vec\mu_1,\vec\lambda_1,Q):\text{ there exist }\mu_2,\lambda_2\in\mathbb C^*\text{ with }(\mu_2,\lambda_2,Q)\in V_{K_n}(1)\text{ and }(\vec\mu_1,\mu_2,\vec\lambda_1,\lambda_2,Q)\in V_K(n)\}.$$

This is a proposed geometric compatibility under removing a link component; the source gives no resolution status.