Degree and leading-term conjecture for normal zeta functions of nilpotent Lie rings

Let $L$ be a class-$c$-nilpotent Lie ring of rank $n$, with upper central series $(Z_i(L))_{i=0}^c$. Set $n_i:=\operatorname{rk}(L/Z_i(L))$, so that $n_0=n=\operatorname{rk}(L)$. Let $\zeta^{\triangleleft}_{L,p}(s)$ denote the local normal-subring zeta function. Degree and leading-term conjecture. For almost all primes $p$,

$$\operatorname{deg}_{p^{-s}}(\zeta^{\triangleleft}_{L,p}(s))=-\sum_{i=0}^{c}n_i,$$ $$\lim_{s\rightarrow -\infty}(p^{-s})^{\sum_{i=1}^{c}n_i}\zeta^{\triangleleft}_{L,p}(s)=(-1)^np^{\binom{n}{2}}.$$

The assertions follow from the relevant functional equation whenever that equation holds; in particular, the conjecture is known for nilpotency class at most $2$. For higher classes the functional equation can fail, although all known examples still satisfy the two displayed assertions.