The categorical McKay correspondence conjecture for finite simply-laced Dynkin diagrams

Let $I$ be a simply-laced Dynkin diagram with Coxeter number $h$. A triangulated category $\mathcal{D}$ is equipped with an exact twist functor $\mathcal{F}\mapsto\mathcal{F}(-2)$.

Categorical McKay correspondence conjecture. There exists such a category and functor satisfying all of the following: $\mathcal{D}$ is 2-periodic, with $T^2=\operatorname{id}$; every object has a canonical functorial isomorphism $\mathcal{F}(2h)\cong\mathcal{F}$; and, for the Grothendieck group $K$ equipped with the inner product defined in the source, the indecomposable objects form a simply-laced root system whose Coxeter element is $C\colon[\mathcal{F}]\mapsto[\mathcal{F}(-2)]$. Choosing a parity function $p\colon I\to\mathbb{Z}_2$ with $p(i)=p(j)+1$ on adjacent vertices, the indecomposables are naturally bijective with

$$\widehat{I}=\{(i,n)\in I\times\mathbb{Z}_{2h}\mid p(i)+n\equiv0\pmod 2\},$$

with $C(i,n)=(i,n-2)$, and are denoted $X_i(n)$. Their Hom and Ext groups, path-algebra description, and the equivalences $R\Phi_h\colon\mathcal{D}\to D^b(I,\Omega_h^{op})/T^2$ for every height function $h\colon I\to\mathbb{Z}_{2h}$ satisfy precisely properties (5) and (6) of the conjecture in the source.

This conjecture proposes a finite-type analogue of the categorical construction of affine root systems arising from the McKay correspondence. It predicts a category whose indecomposable objects encode the root system and whose twist realizes the Coxeter element; the source provides the detailed Hom, Ext, quiver, and derived-equivalence requirements but does not state a resolution.