Functional equation conjecture for overlap type zeta functions of free class-2-nilpotent groups

Let $d\geq 2$, let $d'$ denote the parameter used to define the overlap types, and let $w\in{\mathcal{D}}_{2d'}$. Let $\mathfrak{f}_{2,d}(\mathfrak{o})$ be the relevant free class-$2$-nilpotent object, and write $\zeta^{w}_{\mathfrak{f}_{2,d}}(q,t)$ for its overlap type zeta function. Functional equation conjecture. For all $d\geq 2$ and all $w\in{\mathcal{D}}_{2d'}$, one has

$$\zeta^{w}_{\mathfrak{f}_{2,d}}(q^{-1},t^{-1})=(-1)^Dq^{\binom{D}{2}}t^D\zeta^{w}_{\mathfrak{f}_{2,d}}(q,t).$$

The paper establishes this functional equation for the type $w=\mathsf{0}^{d'}\mathsf{1}^{d'}$ and reports explicit calculations confirming the broader assertion for $d\leq 4$. The conjecture proposes that the same symmetry holds for every overlap type in every dimension covered by the stated range.