k1-injectivity of the Paschke dual algebra for certain simple C*-algebras

Let $\mathcal{B}$ be a nonunital separable simple stable C*-algebra with strict comparison of positive elements and $T(\mathcal{B})$ having finite extreme boundary, and let $\mathcal{A}$ be a simple unital separable nuclear C*-algebra. We prove that the Paschke dual algebra ${\mathcal{A}}_{\mathcal{B}}^d$ is $K_1$-injective. As a consequence, we obtain interesting $KK$-uniqueness theorems which generalize the Brown-Douglas-Fillmore essential codimension property.