Bijective rigidity of uniform Roe algebras and injectivity of the comparison map

We show that, for uniformly locally finite metric spaces $X$ and $Y$ with isomorphic uniform Roe algebras $C_u^{\ast }(X)$ and $C_u^{\ast }(Y)$, the existence of a bijective coarse equivalence $f\text{\hspace{0.17em}⁣}:X\to Y$ is equivalent to the injectivity of the $0$th comparison map appearing in the HK conjecture for coarse groupoids. We further prove that the $0$th comparison map is injective unconditionally. Moreover, if the underlying space is coarsely connected, this map is in fact split-injective.