Linearity and parity conjecture for Rouquier complexes of Mikado braids

Let $({\mathcal{W}},{\mathcal{S}})$ be a Coxeter system with length function $\ell$, let $A\in\mathscr{B}(\Phi^+)$, and let $w_A$ be the associated Mikado braid. Set $\beta:=w_A$, and let $F_\beta$ be a minimal Rouquier complex for $\beta$. Write its cohomological term in degree $i$ as ${}^iF_\beta$, and let $B_z(i)$ denote the grading shift of $B_z$. Linearity and parity conjecture. The following hold: (1) ${}^0F_\beta$ has a unique summand isomorphic to $B_w$, and every other summand is isomorphic to $B_z$ with $z<w$; (2) for every $i\in\mathbb{Z}\backslash\{0\}$, ${}^iF_\beta=\bigoplus B_z(i)^{\oplus m_{z,i}}$ for $z<w$ and $m_{z,i}\in\mathbb{Z}_{\geq0}$, hence $F_\beta\in K^b(\mathcal{B})^{\leq0}\cap K^b(\mathcal{B})^{\geq0}$; and (3) $m_{z,i}=0$ whenever $i$ and $\ell(z)-\ell(w)$ have different parity. This conjecture predicts that minimal Rouquier complexes associated with general Mikado braids have the same linearity, triangularity, and parity properties already established in the special cases treated in the paper.