The ABP conjecture for periodic cyclic homology

Let ${\mathcal G}$ be a reductive $p$-adic group and let ${\mathfrak s}$ be a point in the Bernstein spectrum of ${\mathcal G}$. Let ${\mathcal I}^{{\mathfrak s}}\subset{ \mathcal H}({\mathcal G})$ be the Bernstein ideal labelled by ${\mathfrak s}$, and let $T^{{\mathfrak s}}$ and $W^{{\mathfrak s}}$ be the complex torus and finite group associated to ${\mathfrak s}$. The ABP periodic-cyclic-homology conjecture. There are canonical isomorphisms of complex vector spaces

$${\rm HP}_j({\mathcal I}^{{\mathfrak s}})\simeq\bigoplus_l {\rm H}^{j+2l}(T^{{\mathfrak s}}{/\!/}W^{{\mathfrak s}};\mathbb{C}),$$

for $j=0,1$. This formulation relates the periodic cyclic homology of Bernstein ideals to the orbifold cohomology of the corresponding extended quotient. The source presents it as the periodic-cyclic-homology level of the ABP conjecture; no resolution beyond the stated principal-series results is supplied here.