Reduced zeta-function conjecture for graded ideals in free nilpotent Lie rings

Assume the uniformity conjecture, and define the reduced graded ideal zeta function by

$$\zeta_{\mathfrak{f}_{c,d},\operatorname{red}}^{\triangleleft_{\operatorname{gr}}}(Y)=W_{c,d}^{\triangleleft_{\operatorname{gr}}}(1,Y)\in\mathbb{Q}(Y).$$

Let $r=\sum_{i=1}^cW_d(i)$ and $[N]_0=\{0,\ldots,N\}$. Reduced zeta-function conjecture. The function $\zeta_{\mathfrak{f}_{c,d},\operatorname{red}}^{\triangleleft_{\operatorname{gr}}}(Y)$ has a pole of order $r$ at $1$, and there exist $e_i\in\mathbb{N}$ for $i\in[r]$ and nonnegative integers $a_j\in\mathbb{N}_0$ for $j\in[N]_0$, where

$$N=\left(\sum_{i=1}^re_i\right)-\sum_{j=1}^c(c+1-j)W_d(j),$$

with $a_0=1$, $a_j=a_{N-j}$ for all $j\in[N]_0$, and

$$\zeta_{\mathfrak{f}_{c,d},\operatorname{red}}^{\triangleleft_{\operatorname{gr}}}(Y)=\frac{\sum_{j=0}^Na_jY^j}{\prod_{i=1}^r(1-Y^{e_i})}.$$

The conjecture links these reduced zeta functions to Hilbert--Poincaré series of Cohen--Macaulay and Gorenstein algebras; it is known for $c=2$ by the argument described in the paper.