Cohomological classification conjecture for 2-representations of crossed modules

Let $\mathcal{K}$ be a crossed module with finite fundamental group $\pi_1(\mathcal{K})$, let $\widetilde{\mathcal{K}}$ denote its associated 2-group, and let $\mathcal{X}$ be a classifying space of $\mathcal{K}$. A finite $\pi_1(\mathcal{K})$-set $X$ determines the permutation local system $\underline{({\mathbb K}^{\times})^X}$ on $\mathcal{X}$. Cohomological classification conjecture. 2-representations of $\widetilde{\mathcal{K}}$ are classified by pairs $(X,[\mu])$, where $X$ is a finite $\pi_1(\mathcal{K})$-set and

$$[\mu]\in H^2\bigl(\mathcal{X},\underline{({\mathbb K}^{\times})^X}\bigr).$$

This would place 2-representations in a cohomological framework analogous to the classification of projective representations; the paper presents the associated local system as expected to play a crucial role, but supplies no resolution of the classification claim.