The natural measure conjecture for strongly regular Poisson manifolds

Let $P$ be a strongly regular Poisson manifold whose simply connected symplectic leaves are the fibres of a smooth fibration $P\to M$. Assume that the lattice of differentials of periods is a full lattice in each cotangent space, so that it defines an integer affine structure on $M$. Let $G$ be the symplectic groupoid of $P$, let $Q_A$ be the relevant measure line, and let $\lambda$ be the square of an invariant measure on $P$ factored as the product of Liouville measure along the symplectic leaves and the pull-back of a measure $\beta$ on $M$. Natural measure conjecture. The induced measure on the stack $P//G$ agrees with $\beta$ if and only if $\beta$ is the measure associated to the integer affine structure on $M$. This identifies the natural transverse measure on the leaf space with the measure determined by its integral affine structure, extending the analogy with Liouville measure on symplectic manifolds; the supplied text does not indicate whether the assertion has been proved.