Degree and limiting-value conjecture for submodule zeta functions of nilpotent endomorphism algebras

Let $({\mathcal{L}},{\mathcal{E}})$ be a nilpotent algebra of endomorphisms as above, with associated integers $c$, $n$, and $N_0,\ldots,N_{c-1}$. For almost all prime ideals $\mathfrak{p}$ of ${\mathcal{O}}$ and all finite extensions $\mathfrak O$ of ${\mathcal{O}}_{\mathfrak{p}}$ with residue field cardinality $q^f$, the local submodule zeta function satisfies

$$\deg_{q^{-fs}}\left(\zeta_{{\mathcal{E}}(\mathfrak O)\curvearrowright {\mathcal{L}}(\mathfrak O)}(s)\right)=-\sum_{i=0}^{c-1}N_i,$$

and

$$\lim_{s\rightarrow-\infty}\left(q^{-fs\sum_{i=0}^{c-1}N_i}\zeta_{{\mathcal{E}}(\mathfrak O)\curvearrowright {\mathcal{L}}(\mathfrak O)}(s)\right)=(-1)^nq^{-f\binom{n}{2}}.$$

Degree and limiting-value conjecture. If $({\mathcal{L}},{\mathcal{E}})$ is almost uniform, so that $\zeta_{{\mathcal{E}}(\mathfrak O)\curvearrowright {\mathcal{L}}(\mathfrak O)}(s)=W(q^f,q^{-fs})$ for almost all $\mathfrak p$ and all $\mathfrak O$, for some $W\in{\mathbb Q}(X,Y)$, then $\deg_XW=-\binom{n}{2}$.

These assertions describe the degree and asymptotic behaviour of the local submodule zeta functions, and in the almost uniform case impose a corresponding degree condition on the uniformizing rational function. The supplied text does not identify the statement as proved or provide evidence resolving it.