The smooth homotopy hypothesis for L-infinity algebroids

Let $\mathfrak{g}$ be an $L_\infty$-algebroid and let $\Pi_{\infty}(\mathfrak{g})$ denote its source infinity-connected integration. Let $G$ be a Lie infinity-groupoid integrating $\mathfrak{g}$, and suppose that $G$ is source $n$-connected.

Higher integration conjecture. In the category of $L_\infty$-algebroids over Lie infinity-groupoids, $\mathfrak{g}$ is Morita equivalent to $\Pi_{\infty}(\mathfrak{g})$. More generally, $\mathfrak{g}$ is $n$-equivalent to $G$.

This is presented as a generalization of the smooth homotopy hypothesis and as a statement for which a special case has been proved. The full higher claim remains open.