Compatibility and limiting local cohomology conjectures for arc algebras

Let $J[0^N,(1)^{N'}]$ and $A[0^N,(1)^{N'}]$ be the finite arc algebras, let $\mathfrak{a}$ be the local-cohomology support ideal, and let $\mathsf{th}$ denote the homomorphisms induced by the conjectural constructions in the source. Semi-infinite local-cohomology conjecture. The groups $H_{\mathfrak{a}}^i(J[0^N,(1)^{N'}])$ for different $N,N'$ are related by homomorphisms $\mathsf{th}$ as specified in the source; they satisfy the corresponding compatibility relation needed to define the limiting groups $H_{\mathfrak{a}}^{i+\frac{\infty}{2}}(A_f)$; and the map

$$H_{\mathfrak{a}}^{i+\frac{\infty}{2}}(A)\to H_{\mathfrak{a}}^{i+\frac{\infty}{2}}(A_f)$$

extends the finite-level map. These assertions are intended to construct a well-defined semi-infinite local-cohomology theory from the finite approximations.