The homological characterization of one-way functions

Let $f$ be a function, let

$$L_f=\{(y,x):f(x)=y\},$$

and let $h(L_f)$ denote the homological complexity of the inversion problem associated with $f$. Homological characterization of one-way functions. The function $f$ is one-way if and only if the inversion problem has non-trivial higher homology, equivalently,

$$f\text{ is one-way}\quad\Longleftrightarrow\quad h(L_f)>0.$$

This conjectures a topological characterization of cryptographic one-wayness. The source cites a homological lower-bound theorem for one direction but identifies the converse as requiring a new construction, so the equivalence remains open.