Lusztig's P1–P15 conjectures for Coxeter groups with complete graph

Let $(W,S)$ be a Coxeter group with weight function $L$, Hecke algebra structure constants $h_{x,y,z}$, Kazhdan–Lusztig $\mathbf{a}$-function, and sets $\mathcal{D}_{\geq N}$ and $W_{\geq N}$ defined by the preceding notation. For $N\in\mathbb{N}$, write $\mathcal{D}_{\geq N}=\{z\in\mathcal{D}\mid\mathbf{a}(z)\geq N\}$ and $W_{\geq N}=\{w\in W\mid\mathbf{a}(w)\geq N\}$. Lusztig's P1–P15 conjectures. For every $N\in\mathbb{N}$, the following assertions hold: $(\mathrm{P1})_{\geq N}$: for every $w\in W_{\geq N}$, $\mathbf{a}(w)\leq\Delta(w)$; $(\mathrm{P2})_{\geq N}$: if $z\in\mathcal{D}_{\geq N}$ and $\gamma_{x,y,z}\neq0$, then $x=y^{-1}$; $(\mathrm{P3})_{\geq N}$: if $y\in W_{\geq N}$, there is a unique $z\in\mathcal{D}$ with $\gamma_{y^{-1},y,z}\neq0$; $(\mathrm{P4})_{\geq N}$: if $w'\prec_{LR}w$ and $w\in W_{\geq N}$, then $\mathbf{a}(w')\geq\mathbf{a}(w)$; $(\mathrm{P5})_{\geq N}$: if $z\in\mathcal{D}_{\geq N}$, $\gamma_{y^{-1},y,z}\neq0$, then $\gamma_{y^{-1},y,z}=n_z=\pm1$; $(\mathrm{P6})_{\geq N}$: every $z\in\mathcal{D}_{\geq N}$ satisfies $z^2=e$; $(\mathrm{P7})_{\geq N}$: if one of $x,y,z$ lies in $W_{\geq N}$, then $\gamma_{x,y,z}=\gamma_{y,z,x}=\gamma_{z,x,y}$; $(\mathrm{P8})_{\geq N}$: if one of $x,y,z$ lies in $W_{\geq N}$ and $\gamma_{x,y,z}\neq0$, then $x\sim_Ly^{-1}$, $y\sim_Lz^{-1}$, and $z\sim_Lx^{-1}$; $(\mathrm{P9})_{\geq N}$: if $w'\prec_Lw$, $w\in W_{\geq N}$, and $\mathbf{a}(w')=\mathbf{a}(w)$, then $w'\sim_Lw$; $(\mathrm{P10})_{\geq N}$: the analogous assertion holds for the right preorder and right-cell equivalence; $(\mathrm{P11})_{\geq N}$: the analogous assertion holds for the two-sided preorder and two-sided-cell equivalence; $(\mathrm{P12})_{\geq N}$: for $I\subseteq S$ and $y\in W_I\cap W_{\geq N}$, the $\mathbf{a}$-value of $y$ in $W_I$ equals its value in $W$; $(\mathrm{P13})_{\geq N}$: every left cell $\Gamma\subseteq W_{\geq N}$ contains a unique $z\in\mathcal{D}$, and $\gamma_{y^{-1},y,z}\neq0$ for every $y\in\Gamma$; $(\mathrm{P14})_{\geq N}$: every $w\in W_{\geq N}$ satisfies $w\sim_{LR}w^{-1}$; and $(\mathrm{P15})_{\geq N}$: for $w,w'\in W$ and $x,y\in W_{\geq N}$ with $\mathbf{a}(x)=\mathbf{a}(y)$,

$$\sum_{z\in W}h_{w,x,z}\otimes h_{z,w',y}=\sum_{z\in W}h_{w,z,y}\otimes h_{x,w',z}\in\mathcal{A}\otimes_{\mathbb{Z}}\mathcal{A}.$$

These are the truncated forms of Lusztig's fifteen conjectures governing the $\mathbf{a}$-function, distinguished elements, leading coefficients, and cell preorders in Coxeter-group Hecke algebras. The supplied text does not state whether this full collection is open or resolved, so the database status remains open.