Dual Log-compatibility conjecture for multiplicative quiver-stack point counts

Let $Q=(I,\Omega)$ be a quiver, let $\mathcal{M}_{s,\alpha}$ be the associated multiplicative quiver stack over $\mathbb{F}_q$, and define the class function $c_{\alpha}:\operatorname{GL}_{\alpha}(\mathbb{F}_q)\to\mathbb{C}$ by

$$c_{\alpha}(g)=\dfrac{|(\mu^{\circ}_{\alpha})^{-1}(g)^F|}{q^{-(\alpha,\alpha)}}.$$

A family of class functions is dual Log compatible in the sense defined earlier in the paper. Dual Log-compatibility conjecture. The family of class functions $\{c_{\alpha}\}_{\alpha\in\mathbb{N}^I}$ is dual Log compatible. The conjecture is presented as an expected technical property needed to control rational-point formulas for multiplicative quiver stacks; no resolution is given in the supplied text.