Uniform symmetry-factor conjecture for pro-isomorphic zeta functions of graded Lie lattices

Let $\mathcal{L}$ be a graded Lie lattice. A grading is a decomposition of the underlying free $\mathbb{Z}$-module compatible with the Lie bracket, and its weight is the corresponding weighted rank; write $\mathrm{wt}(\mathcal{L})$ for the minimum of these weights. Let $\zeta^\wedge_{\mathcal{L},p}(s)$ denote the local pro-isomorphic zeta function. Suppose that there are rational functions $W_1,\dots,W_k\in\mathbb{Q}(X,Y)$ such that, for almost all primes $p$, one of them satisfies $\zeta^\wedge_{\mathcal{L},p}(s)=W_i(p,p^{-s})$. Suppose further that there exist $a,b,c\in\mathbb{Z}$ such that

$$W_i(X^{-1},Y^{-1})=(-1)^c p^{a-bs}W_i(X,Y)$$

for every $i$. Uniform symmetry-factor conjecture. Then $b=\mathrm{wt}(\mathcal{L})$. The conjecture predicts that the exponent governing the uniform functional-equation symmetry factor is determined by the minimal grading weight. Functional equations are known in several classes of Lie lattices, but the asserted identification of this exponent with the minimal weight is not established in general.