Reconstruction conjecture for commutative measured quantum groupoids

Let $(N,M,\alpha,\beta,\Gamma,\mu,T_L,T_R)$ be a measured quantum groupoid, and suppose that $M$ is commutative. Let $G$ denote a locally compact groupoid, with unit space $G^{\{0\}}$, Haar-system measure $\nu$, coproduct $\Gamma_G$, invariant operator-valued weight $P_G$, and involution $j_G$. Reconstruction conjecture. There exists a locally compact groupoid $G$ such that

$$(N,M,\alpha,\beta,\Gamma,\mu,T_L,T_R)\simeq (L^{\infty}(G^{\{0\}},\mu),L^{\infty}(G,\nu),\alpha,\beta,\Gamma_G,\mu,P_G,j_G\circ P_G\circ j_G).$$

This asserts that every measured quantum groupoid with commutative von Neumann algebra $M$ arises from a locally compact groupoid. The surrounding results construct the commutative measured quantum groupoid associated with a measured groupoid, but the converse reconstruction claim is not resolved in the supplied text.