  In this paper we classify the irreducible Harish-Chandra bimodules with full
support over filtered quantizations of conical symplectic singularities under
the condition that none of the slices to codimension 2 symplectic leaves has
type $E_8$. More precisely, we show that the top quotient
$\overline{\operatorname{HC}}(\mathcal{A}_\lambda)$ of the category of
Harish-Chandra bimodules over the quantization $\mathcal{A}_\lambda$ with
parameter $\lambda$ embeds into the category of representations of the
algebraic fundamental group, $\Gamma$, of the open leaf. The image coincides
with the representations of $\Gamma/\Gamma_\lambda$, where $\Gamma_\lambda$ is
a normal subgroup of $\Gamma$ that can be recovered from the quantization
parameter $\lambda$. As an application of our results, we describe the Lusztig
quotient group in terms of the geometry of the normalization of the orbit
closure in almost all cases.
