When does the Thom class lift through the coarse co-assembly map away from a submanifold?

Determine conditions under which the Thom class of a submanifold N with K-oriented normal bundle, viewed as an element of K(A(N \Subset M; Cl_{0,r})), lies in the image of the coarse co-assembly map away from N, namely \mu_N: K(sHigCor_N(M; Cl_{1,r})) → K((N \Subset M; Cl_{0,r})).

Background

The authors build an abstract machinery producing wrong-way maps on K-theory of Roe algebras, mapping the coarse index on M to the index on a suitably embedded submanifold N. This machinery hinges on representing the Thom class of the normal bundle via a class in the stable Higson corona and lifting it through a coarse co-assembly map away from N. They show this can be done for multi-partitioned manifolds but seek general criteria ensuring such a lift exists, which would yield broad submanifold obstructions to uniform positive scalar curvature away from N.