Injectivity conjecture for the proposed BPS Lie algebra generators

Let $J$ and $L^{\mathcal{S}}$ be the morphisms constructed from the proposed generators of the BPS Lie algebra associated with the preprojective CoHA, where $\mathcal{S}$ is a Serre subcategory. Injectivity conjecture. The morphisms $J$ and $L^{\mathcal{S}}$ are injective. The source calls this a conjecture and gives no evidence of a resolution.