The BPS Lie algebra conjecture for symmetric quivers with potential

Let $Q$ be a symmetric quiver with potential $\mathsf{W}$. Let $\mathfrak{g}_{Q,\mathsf{W}}$ be the Lie superalgebra defined from the classical $R$-matrix, and write

$$\mathfrak{g}^+_{Q,\mathsf{W}}=\bigoplus_{\eta>0}\mathfrak{g}_{\eta}.$$

The positive part of $\mathfrak{g}_{Q,\mathsf{W}}$ should be identified with the BPS Lie algebra of Davison--Meinhardt. BPS Lie algebra conjecture. There is a $\mathbb{C}(\mathsf t_0)$-Lie algebra isomorphism

$$\mathfrak{g}^+_{Q,\mathsf{W}}\cong \mathfrak{g}^{\mathrm{BPS},\mathsf T_0}_{Q,\mathsf{W}}\otimes_{\mathbb{C}[\mathsf t_0]}\mathbb{C}(\mathsf t_0)$$

that intertwines their actions on any state space $\mathcal{H}^{\mathsf{W}^{\mathrm{fr}}}_{\underline{\mathbf{d}},\mathsf A^{\mathrm{fr}}}$. This is the proposed comparison between the classical $R$-matrix Lie algebra and the lowest perverse component of the CoHA; the analogous identification for tripled quivers with canonical cubic potential is known, while the general case remains open.