Integral cocenter isomorphism conjecture for the Gieseker variety

Let $\mathcal{R}$ be the coefficient ring, let $P_{\mathrm{loc}}=P\otimes_{\mathcal{R}}\operatorname{Frac}\mathcal{R}$ for every $\mathcal{R}$-module $P$, and let

$$K_{\mathcal{T}}(\mathcal{L}(n,r))_{\mathrm{loc}}\simeq \operatorname{Tr}({\bf H}_{n,r})_{\mathrm{loc}}$$

be the unique isomorphism of $Z({\bf H}_{n,r})_{\mathrm{loc}}$-modules intertwining the specified pairings and maps. Integral cocenter isomorphism conjecture. This isomorphism induces the isomorphism

$$K_{\mathcal{T}}(\mathcal{L}(n,r))\simeq \operatorname{Tr}(\Heck_{n,r})$$

of $Z(\Heck_{n,r})$-modules. The conjecture asserts that the localized identification descends to the unlocalized equivariant $K$-theory and cocenter; its resolution is not specified in the supplied text.