The p-adic residue conjecture for nilpotent algebras

Let $\cA$ be a nilpotent $\fo$-algebra that is free of rank $d$ as an $\fo$-module, where $\fo$ is the ring of integers in a number field $k$. Let $K\supset k$ be a $p$-adic field with valuation ring $\fO$. $p$-adic residue conjecture. The local zeta function $\zeta_{\cA_{\fO}}(s)$ has a simple pole at zero and

$$\left.\frac{\zeta_{\cA_{\fO}}(s)}{\zeta_{(\fO^d,0)}(s)}\right|_{s=0}=1.$$

This is presented as a possible $p$-adic shadow of the topological residue conjecture and is noted as a conjecture not previously recorded by the author.