Brick-sequence exact-sequence conjecture

Let $W$ be a Weyl group, let $w\in W$, and let $B_1,\dots,B_l$ be a brick sequence of the torsion-free class $\mathcal{F}(w)$. Suppose that

$$\operatorname{\underline{\dim}} B_m=\operatorname{\underline{\dim}} B_i+\operatorname{\underline{\dim}} B_j$$

for $1\leq i<m<j\leq l$. Brick-sequence exact-sequence conjecture. There is an exact sequence

$$0\longrightarrow B_i\longrightarrow B_m\longrightarrow B_j\longrightarrow 0.$$

The conjecture strengthens the construction used in the proof of the paper's main theorem, where only a nonzero non-injective morphism $B_m\to B_j$ is obtained. It generalizes the corresponding result for path algebras, proved over an algebraically closed field.