Equality of relative coarse index constructions (this paper vs Roe)

Establish whether the relative coarse index defined in Roe’s work (Roe, “Index Theory, Coarse Geometry, and Topology of Manifolds,” Section 4) agrees, up to canonical maps, with the relative coarse index constructed in this paper via gluing of Dirac data along codimension-zero submanifolds with boundary.

Background

The paper develops a coarse version of the relative index and notes that Roe already defined a relative coarse index using different methods. Although the authors prove a relative index theorem analogous to both Gromov–Lawson’s classical result and Roe’s theorem, they have not verified the equivalence of the two constructions. Demonstrating equality (up to canonical maps) would unify the approaches and clarify functorial relations among the various relative index frameworks.