Postnikov classification conjecture for semi-weak monoidal 2-categories

Let $G$ be a group, $H$ an abelian group, and $R$ a commutative ring with unit group $R^*$. Let $B_G$ be the classifying space of $G$, and for each cohomology class $\alpha\in H^3(B_G,H)$ let $W(\alpha)$ be a CW-approximation of the principal fibration induced by a representative map $B_G\to B_H^3$. Consider the semi-weak monoidal 2-category structures on ${\mathbb N}(G,H,R)$ and their equivalence classes. Postnikov classification conjecture. These equivalence classes correspond bijectively to pairs of cohomology classes

$$\alpha\in H^3(B_G,H),\qquad \beta=\beta_{\alpha}\in H^4(W(\alpha),R^*).$$

This is motivated by the Postnikov classification of connected 3-types with homotopy groups $G$, $H$, and $R^*$, but the correspondence between the categorical structures and these cohomological data remains conjectural.