Main Borcherds-algebra conjecture for BPS Lie algebras

Fix a Serre subcategory $\mathcal{S}$, a stability condition $\upzeta$, and a slope $\uptheta$. Let $\operatorname{\mathcal{S}ta}^{G,\upzeta}_{\Pi_Q,\uptheta}$ be the direct sum of intersection complexes of stable dimension vectors, and let $\operatorname{\mathcal{I}so}^{G,\upzeta}_{\Pi_Q,\uptheta}$ be the direct sum of the diagonal contributions from primitive isotropic dimension vectors. Denote their induced generator objects by $\mathfrak{st}^{\mathcal{S},G,\upzeta}_{\Pi_Q,\uptheta}$ and $\mathfrak{is}^{\mathcal{S},G,\upzeta}_{\Pi_Q,\uptheta}$, respectively; let $\mathcal{BPS}^{G,\upzeta}_{\Pi_Q,\uptheta}$ and $\mathfrak{g}^{\mathcal{S},G,\upzeta}_{\Pi_Q,\uptheta}$ be the corresponding BPS Lie algebra objects. Main Borcherds-algebra conjecture. The inclusions of the stable and isotropic objects extend to isomorphisms

$$\operatorname{Bor}^+\left(\operatorname{\mathcal{S}ta}^{G,\upzeta}_{\Pi_Q,\uptheta}\oplus \operatorname{\mathcal{I}so}^{G,\upzeta}_{\Pi_Q,\uptheta}\right)\cong \mathcal{BPS}^{G,\upzeta}_{\Pi_Q,\uptheta}$$

and

$$\operatorname{Bor}^+\left(\mathfrak{st}^{\mathcal{S},G,\upzeta}_{\Pi_Q,\uptheta}\oplus \mathfrak{is}^{\mathcal{S},G,\upzeta}_{\Pi_Q,\uptheta}\right)\cong \mathfrak{g}^{\mathcal{S},G,\upzeta}_{\Pi_Q,\uptheta}.$$

This is the paper's main conjecture: the displayed stable and isotropic objects should provide all BPS generators, while the source gives no proof.