Define the balanced part along mutation sequences for general n>2

Develop a definition of the balanced subalgebra $\mathcal Z_{\hat\omega}^{\rm bl}(\mathcal D_j)$ for each intermediate mutated seed $\mathcal D_j$ with $1\leq j\leq r-1$ in the quantum coordinate change framework for ${\rm SL}_n$ with $n>2$, so that a balanced part is available at every step of the mutation sequence rather than relying on the mutable-balanced subalgebra $\mathcal Z^{\rm mbl}$.

Background

To compare representations across triangulations, the paper uses balanced quantum coordinate changes and constructs representations via the balanced Fock–Goncharov algebra. For $n=2$, a single mutation suffices and the balanced part can be handled directly.

For $n>2$, the authors state they do not know how to define the balanced part $\mathcal Z_{\hat\omega}^{\rm bl}(\mathcal D_j)$ at intermediate stages of a mutation sequence, necessitating the use of the mutable-balanced subalgebra $\mathcal Z^{\rm mbl}$. A definition of the balanced part for all intermediate seeds would strengthen the naturality and conceptual clarity of the framework.