Vogan's augmented Kazhdan–Lusztig hypothesis for p-adic groups

Let $G$ be a connected reductive algebraic group over a locally compact non-Archimedean local field $F$. Let $\lambda:W_F\to{}^LG$ be an infinitesimal parameter. For geometric parameters $\gamma$ and $\xi$, write $m_r[\gamma,\xi]$ for the multiplicity of the representation $\pi_\gamma$ in the standard representation $S_\xi$, let $d_\gamma=\dim\mathcal{O}_\gamma$, and let $c_g$ and $\hat{c}_g$ be the geometric character matrices defined by $c_g[\gamma,\xi]=(-1)^{d_\gamma}\chi_\gamma(P_\xi)$ and $\hat{c}_g[\gamma,\xi]=(-1)^{d_\gamma}c_g[\gamma,\xi]$. Vogan's augmented Kazhdan–Lusztig hypothesis. For all $\gamma$ and $\xi$,

$$m_r[\gamma, \xi] = (-1)^{d_\gamma+d_\xi} c_g[\xi, \gamma]=(-1)^{d_\gamma}\hat{c}_g[\xi, \gamma].$$

The original $p$-adic Kazhdan–Lusztig hypothesis has a different sign convention, but the displayed correction is motivated by an example in which the original matrix has negative entries and therefore cannot give multiplicities of irreducible representations in standard representations. The conjecture is presented here without a resolution status.