The ultimate homological characterization of complexity classes

Let $\mathcal{C}$ be a natural complexity class, let $h(L)$ denote the homological complexity of a problem $L$, and let $S_{\mathcal{C}}$ be a subset of $\mathbb{N}\cup\{\infty\}$. Ultimate homological characterization. Every natural complexity class $\mathcal{C}$ can be characterized by

$$\mathcal{C}=\{L:h(L)\in S_{\mathcal{C}}\}$$

for some set $S_{\mathcal{C}}\subseteq\mathbb{N}\cup\{\infty\}$. This conjectures that homological complexity completely classifies natural complexity classes. The source gives no resolution and does not specify which sets $S_{\mathcal{C}}$ arise, so the claim remains open.