Cohomological invariance under Lie algebroid n-equivalence

Let $f:A\to B$ be a map of Lie algebroids satisfying the hypotheses of the definition of an $n$-equivalence. Let $H^*(A,\mathcal{O})$ and $H^*(B,\mathcal{O})$ denote the corresponding Lie algebroid cohomology groups with coefficients in $\mathcal{O}$.

Cohomological invariance conjecture. One has

$$H^*(A,\mathcal{O})=H^*(B,\mathcal{O}).$$

The claim predicts invariance of Lie algebroid cohomology under the specified class of maps and is intended to support the proposed homotopy-theoretic localization at $n$-equivalences. It remains conjectural in the source.