The coherent Springer sheaf heart conjecture

Let $G$ be the reductive group under consideration, let $\widetilde{G}$ denote its Grothendieck resolution, and let ${\mathcal N}$ be the nilpotent cone in its Lie algebra. Write $\mathcal L({\mathcal N}/\widetilde{G})$ for the derived loop space and ${\mathcal S}$ for the coherent Springer sheaf. Coherent Springer sheaf heart conjecture. The Springer sheaf ${\mathcal S}$ lives in the abelian category

$${\operatorname{Coh}}({\mathcal L}({\mathcal N}/\widetilde{G}))^\heartsuit.$$

This asserts that the coherent Springer sheaf is a sheaf rather than a genuinely derived coherent object. The paper proves the assertion for $G=GL_2$ and $G=SL_2$, while the general case remains open.