Linear-ordering conjecture for reflection representations of skew-symmetrizable matrices

Let $B$ be a skew-symmetrizable matrix indexed by $\mathcal I={1,\ldots,n}$. For a mutation sequence $\boldsymbol{w}$, let $r_i^{\boldsymbol{w}}$ be the associated reflections, let $C^{\boldsymbol{w}}$ be its $C$-matrix, and let $pi$ be the representation determined by a linear ordering $prec$ and its associated generalized intersection matrix. Linear-ordering conjecture. For any skew-symmetrizable matrix $B$, there exists a linear ordering $prec$ on $\mathcal I$ such that, if $boldsymbol{w}$ and $boldsymbol{v}$ are mutation sequences satisfying

$$C^{boldsymbol{w}}=C^{boldsymbol{v}},$$

then

$$pi(r_i^{boldsymbol{w}})=pi(r_i^{boldsymbol{v}}),\qquad i=1,\ldots,n.$$

In particular, $B^{boldsymbol{w}}=B^{boldsymbol{v}}$. The conjecture asks whether one ordering makes the represented reflections depend only on the resulting $C$-matrix; the paper investigates this problem in the setting of quiver mutations, while its general status is not resolved in the supplied text.