Invariance of representations up to homotopy under Lie algebroid n-equivalence

Let $A\to X$ and $B\to Y$ be $n$-equivalent Lie algebroids. A representation up to homotopy of length at most $n$ is a representation up to homotopy whose length is bounded by $n$.

Invariance conjecture. There is an equivalence of categories between representations up to homotopy of length at most $n$ of $A$ and of $B$.

This is part of the proposed generalized homotopy theory for Lie algebroids, where $n$-equivalences are intended to preserve structures visible through degree or length $n$. The claim is presented as conjectural and no resolution is supplied.