Rank-two discriminant multiplicity theorem

Let $W\subset L^\vee_\mathbb R$ be a wall, let $\Gamma\subset\Pi$ be a minimal face, and let $H=H_{\Gamma^c}$ be the corresponding relevant subspace. Assume that

$$H\subseteq\langle W\rangle.$$

Write $n_{\Gamma,W}$ for the multiplicity of $D^b(Z_H)$ in $D^b(Z_W)$, and $m_{\Gamma,W}$ for the intersection multiplicity of $\overline{\nabla}_\Gamma$ with $C_W$. Rank-two discriminant multiplicity theorem. If the torus $T$ has rank $2$, then

$$n_{\Gamma,W}=m_{\Gamma,W}.$$

This proves the paper's multiplicity conjecture in rank two; the corresponding equality in higher rank is not established here.