The precise homological complexity characterization of complexity classes

Let $\mathcal{C}$ be a natural complexity class, let $h(L_n)$ denote the homological complexity of the restriction $L_n$ of a problem $L$ to inputs of size $n$, and let $f_{\mathcal{C}}:\mathbb{N}\to\mathbb{N}$. Precise homological complexity characterization. For every natural complexity class $\mathcal{C}$, there exists a function $f_{\mathcal{C}}$ such that

$$\mathcal{C}=\{L:h(L_n)\leq f_{\mathcal{C}}(n)\text{ for all }n\}.$$

This conjectures a size-sensitive homological characterization of every natural complexity class. The source presents it as a future research direction and gives no proof or resolution.