Ruan's orbifold product conjecture for periodic cyclic homology

Let $\mathcal{X}$ be a smooth Deligne--Mumford stack with an étale cover $\pi:U\to\mathcal{X}$ by a scheme, and set $Z=U\times_{\mathcal{X}}U$. The correspondence $Z\times_UZ\to Z$ defines, for $\mathcal{F},\mathcal{G}\in D^b(Z)$, the product

$$\mathcal{F}*\mathcal{G}=m_*\bigl(p_1^*\mathcal{F}\otimes p_2^*\mathcal{G}\bigr).$$

When $\mathcal{X}=[X/G]$ and $U=X$, one has $Z=\coprod_{g\in G}X^g$. The orbifold product conjecture. The product $(\mathcal{F},\mathcal{G})\mapsto\mathcal{F}*\mathcal{G}$ induces the product on the periodic cyclic homology of $Z$, coinciding with the product on

$$A(X,G)=H^*(Z,\mathbb{C}),$$

and passing to $G$-coinvariants or invariants gives the orbifold product on $H^*_{\operatorname{orb}}(X/G,\mathbb{C})$. This proposed compatibility is intended to explain the orbifold product through the derived category and cyclic homology. The source uses expectation language and gives no resolution status.