The cubical complex conjecture for homology of asynchronous transition systems

Let $\Sigma=(S,E,I,\operatorname{Tran})$ be a state space, let $M(E,I)$ be its trace monoid, and let $K(\Sigma)$ be the relevant category. Let $U\circ\Sigma$ be the associated semicubical set, and define its sub-semicubical set $Q'_*(U\circ\Sigma)$ by retaining precisely those cubes whose execution does not reach the sink. For a functor $F:K(\Sigma)\rightarrow Ab$, extend $F$ to $K_*(\Sigma)$ by setting $F(\star)=0$. The complex has degree-$n$ groups

$$\bigoplus_{(s,e_1,\ldots,e_n)\in Q'_n(U\circ\Sigma)}F(s)$$

and differential

$$d_n(s,e_1,\ldots,e_n,f)=\sum_{i=1}^n(-1)^i\left((s\cdot e_i,e_1,\ldots,\widehat{e_i},\ldots,e_n,F(s\stackrel{e_i}{\longrightarrow}s\cdot a_i)(f))-(s,e_1,\ldots,\widehat{e_i},\ldots,e_n,f)\right).$$

Cubical homology conjecture. If $M(E,I)$ is locally finite-dimensional, then for all integers $n\geqslant 0$, the groups $\underrightarrow{\lim}_n^{K(\Sigma)}F$ are isomorphic to the $n$-th homology groups of this complex.

The conjecture is proposed as a way to compute all integer homology groups of finite Petri CE nets, an open problem identified in the cited prior work. The supplied text gives no evidence that this proposed isomorphism has subsequently been proved or disproved.