Bachmann's conjecture on relations among Schlesinger–Zudilin $q$-multiple zeta values

Let $\mathcal{U}$ be the alphabet used to form the noncommutative word algebra $\mathbb{Q}\langle\mathcal{U}\rangle$, let $\mathbb{Q}\langle\mathcal{U}\rangle^\circ$ be the subspace spanned by words not starting with $u_0$, and let

$$\zeta_q^\mathrm{SZ}\colon \mathbb{Q}\langle\mathcal{U}\rangle^\circ\to\mathbb{Q}\llbracket q\rrbracket$$

be the Schlesinger–Zudilin $q$-multiple-zeta-value map, with image $\mathcal{Z}_q$. The algebra carries the stuffle product $\ast$, and it has the involution duality $\tau$ defined on words by

$$\tau\left(u_{k_1}u_0^{z_1}\cdots u_{k_d}u_0^{z_d}\right)=u_{z_d+1}u_0^{k_d-1}\cdots u_{z_1+1}u_0^{k_1-1}.$$

Bachmann's conjecture. All $\mathbb{Q}$-linear relations among elements of $\mathcal{Z}_q$ are generated by the stuffle product $\ast$ and duality $\tau$. This conjecturally gives a complete description of the linear relations among Schlesinger–Zudilin $q$-multiple zeta values, while the general completeness of these two families of relations remains open.