Finite nuclear dimension implies finite complexity relative to subhomogeneous algebras

Let $A$ be a separable $C^*$-algebra. Let $\mathcal{S}$ denote the class of subhomogeneous $C^*$-algebras, namely those $C^*$-algebras that embed into $M_N(C(X))$ for some $N\in\mathbb{N}$ and compact Hausdorff space $X$. Say that $A$ has finite complexity relative to $\mathcal{S}$ when it belongs to a finite complexity level constructed using $\mathcal{S}$ as the base class.

Relative complexity conjecture. Every separable $C^*$-algebra with finite nuclear dimension has finite complexity relative to the class of subhomogeneous $C^*$-algebras.

This is proposed as a counterpart to the real-rank-zero conjecture: dropping real rank zero leads naturally to complexity relative to subhomogeneous algebras, and would provide another route toward understanding the UCT.