GSS uniformity conjecture for free nilpotent Lie rings

Let $\mathfrak{f}_{c,d}$ denote the free class-$c$ nilpotent Lie rings on $d$ generators defined over $\mathbb{Z}$. For $c,d\in\mathbb{N}$, let $\zeta_{\mathfrak{f}_{c,d}}^{\triangleleft}(s)$ be its ideal zeta function. GSS uniformity conjecture. For any $c,d\in\mathbb{N}$, there exists a rational function $W(X,Y)\in\mathbb{Q}(X,Y)$ such that, for almost all primes $p$,

$$\zeta_{\mathfrak{f}_{c,d}(\mathbb{Z}_p)}^{\triangleleft}(s)=W(p,p^{-s}).$$

The conjecture concerns uniformity of local ideal zeta functions as the prime varies. It is known for class $c=2$ and for $\mathfrak{f}_{3,2}$, while the cases such as $\mathfrak{f}_{3,3}$ and $\mathfrak{f}_{4,2}$ remain out of reach.