Bounded geometry Borel conjecture for foliations

Let $(M,\mathcal{F})$ be a foliation on a compact manifold, let $BG$ be the classifying space of its holonomy groupoid, and let $S_x$ be the leaf through $x\in BG$. Let $\mathcal{L}^{uf}$ be the cosheaf of uniformly finite $L$-homology of the leaves, and let $L_*^{\mathcal{F},bg}(M)$ denote the bounded-geometry foliated surgery group. Bounded geometry Borel conjecture for foliations. If the leaves of $(M,\mathcal{F})$ are aspherical, then

$$H_*(BG;\mathcal{L}^{uf}(S_x))\to L_*^{\mathcal{F},bg}(M)$$

is an isomorphism. This is the bounded-geometry Borel-type rigidity statement corresponding to the bounded-geometry foliated surgery theory.