Cohomological invariance under naturally isomorphic Lie groupoid morphisms

Let $f_1:H_1\to G_1$ and $f_2:H_2\to G_2$ be morphisms of Lie groupoids with a natural isomorphism $f_1\Rightarrow f_2$. Let $M_1$ be a $G_1$-module and let $M_2$ be the corresponding $G_2$-module. Write $B^\bullet H_i$ for the nerve of $H_i$, $f_i^{\bullet-1}$ for inverse image of sheaves, and $\mathcal{O}(M_i)$ for the sheaf associated to $M_i$.

Natural-isomorphism invariance conjecture. The natural isomorphism induces an isomorphism

$$H^*(B^\bullet H_1,f_1^{\bullet-1}\mathcal{O}(M_1))\cong H^*(B^\bullet H_2,f_2^{\bullet-1}\mathcal{O}(M_2)).$$

The conjecture formalizes the agreement of the cohomology constructions in the examples preceding it and is intended as a step toward a generalized van Est theorem. It remains open in the source.