The uniform geometric K-homology conjecture

Let $X$ be a metric space. A uniform geometric cycle is a triple $(M,E,f)$ consisting of a spin$^c$ manifold $M$ of bounded geometry, a vector bundle $E\to M$ of bounded geometry, and a suitably uniform map $f:M\to X$. Let $K^{u,\mathrm{geo}}_*(X)$ be the resulting uniform geometric K-homology, and let $K^u_*(X)$ be uniform analytic K-homology.

Uniform geometric K-homology conjecture. The natural map

$$K^{u,\mathrm{geo}}_*(X)\longrightarrow K^u_*(X),\qquad [M,E,f]\longmapsto f_*[\slashed{D}_E]$$

is an isomorphism for reasonable spaces $X$.

This conjecture is the uniform analogue of the identification of geometric and analytic K-homology. The source explains why the map should respect bordism and vector bundle modification but gives no resolution status.