Degree conjecture for topological graded ideal zeta functions

Let $c\in\mathbb{N}$ and $d\in\mathbb{N}_{\geq2}$, let $\mathfrak{f}_{c,d}$ be the free nilpotent Lie ring of class $c$ on $d$ generators, and put $r=\operatorname{rk}_{\mathbb{Z}}(\mathfrak{f}_{c,d})$. Write $\zeta_{\mathfrak{f}_{c,d},\operatorname{top}}^{\triangleleft_{\operatorname{gr}}}(s)$ for its topological graded ideal zeta function. Degree conjecture.

$$\deg_s\left(\zeta_{\mathfrak{f}_{c,d},\operatorname{top}}^{\triangleleft_{\operatorname{gr}}}(s)\right)=-r.$$

This is presented as a graded analogue of Rossmann's Conjecture I; the paper states that the general conjectures are confirmed in relevant special cases, but does not establish this claim in full.