Lambda-independence conjecture for BPS sheaves of deformed preprojective algebras

Let $Q=(Q_0,Q_1)$ be a quiver, let $\lambda\in\mathbf{C}^{Q_0}$, and let $\Pi_Q^{\lambda}$ be the $\lambda$-deformed preprojective algebra. Write $\mathcal{BPS}_{\Pi_Q^{\lambda},\mathrm{Lie}}$ for its BPS sheaf, and let $A_{Q,\mathbf{d}}(t^{-2})$ denote the stated character expression. Lambda-independence conjecture. For any $\lambda,\mu\in\mathbf{C}^{Q_0}$ and $\mathbf{d}\in\mathbf{N}^{Q_0}$ satisfying $\lambda\cdot\mathbf{d}=\mu\cdot\mathbf{d}=0$, there is an isomorphism

$$\mathcal{BPS}_{\Pi_Q^{\lambda},\mathbf{d}}\cong\mathcal{BPS}_{\Pi_Q^{\mu},\mathbf{d}}.$$

In particular, for any $\lambda\in\mathbf{C}^{Q_0}$ and $\mathbf{d}\in\mathbf{N}^{Q_0}$ with $\lambda\cdot\mathbf{d}=0$,

$$\operatorname{ch}\operatorname{H}^*\mathcal{BPS}_{\Pi_Q^{\lambda},\mathbf{d}}=A_{Q,\mathbf{d}}(t^{-2}).$$

Moreover, the BPS cohomology gives a local system on $\{\lambda\in\mathbf{C}^{Q_0}\mid\lambda\cdot\mathbf{d}=0\}$. This generalizes the indivisible-case result for deformed preprojective algebras and is presented as an open strengthening of Kac-type positivity.