The van Est theorem for double Lie groupoids

Let a double Lie groupoid have an associated LA-groupoid obtained by differentiation in the vertical direction. Let the vertical source fiber be $n$-connected, where connectivity is understood in the relevant $(2,1)$-categorical sense.

Double-groupoid van Est conjecture. There is a van Est map from the cohomology of the double Lie groupoid to the cohomology of its LA-groupoid. If differentiation takes place in the vertical direction and the vertical source fiber is $n$-connected, then this map is an isomorphism through degree $n$ and is injective in degree $n+1$.

This is the proposed higher analogue of the ordinary van Est isomorphism theorem. It remains conjectural in the source.