On the small boundary property and $\mathcal{Z}$-absorption

We introduce the Property © for a unital commutative sub-C*-algebra $D$ of a unital C*-algebra $A$, a version of the relative comparison property using almost normalizers. Under the assumption of this property, the $\mathcal{Z}$-absorption of $A$ is shown to imply the small boundary property of $(D,\mathrm{T}(A){\mathrm{\mid }}_D)$, where $A=\mathrm{C}(X)⋊{\mathbb{Z}}^d$ and $D=\mathrm{C}(X)$.