The topological residue conjecture for nilpotent algebras

Let $\cA$ be a Lie algebra or a non-unital associative $\fo$-algebra, free of rank $d$ as an $\fo$-module, where $\fo$ is the ring of integers in a number field $k$, and suppose that $\cA$ is nilpotent. Topological residue conjecture. The function $\zeta_{\cA,\topo}(\ess)$ has a simple pole at zero with residue

$$\frac{(-1)^{d-1}}{(d-1)!}.$$

This strengthens the general pole-at-zero conjecture in the nilpotent Lie and non-unital associative cases, while examples show that assumptions on $\cA$ are necessary and that the pole need not be simple in general.