Topological graded ideal zeta function at zero conjecture

Let $c\in\mathbb{N}$ and $d\in\mathbb{N}_{\geq2}$, let $W=W_d$ be the Witt function, and let $\mathfrak{f}_{c,d}$ be the free nilpotent Lie ring of class $c$ on $d$ generators. Write $\zeta_{\mathfrak{f}_{c,d},\operatorname{top}}^{\triangleleft_{\operatorname{gr}}}(s)$ for its topological graded ideal zeta function. Topological-at-zero conjecture. The function has a pole of order $c$ at $s=0$, with leading coefficient

$$\left.s^c\zeta_{\mathfrak{f}_{c,d},\operatorname{top}}^{\triangleleft_{\operatorname{gr}}}(s)\right|_{s=0}=\frac{(-1)^{\sum_{i=1}^c(W(i)-1)}\prod_{i=1}^cW(i)}{\prod_{i=1}^c\left(\sum_{j=1}^iW(j)\right)(W(i)-1)!}.$$

This is one of the paper's closing conjectures about behaviour at $s=0$ and is supported by the special cases established in the paper.