The bounded-width boundary conjecture for Borel homomorphism problems

A homomorphism problem is a decision problem asking whether a given structure admits a homomorphism to a fixed template. In the Borel setting, its descriptive complexity is measured using the projective classes $\mathbf{\Pi}^1_1$ and $\mathbf{\Sigma}^1_2$; a problem is bounded width when it is solvable by a bounded-width constraint-propagation procedure.

Bounded-width boundary conjecture. A homomorphism problem is $\mathbf{\Pi}^1_1$ iff it is not $\mathbf{\Sigma}^1_2$-complete iff it is bounded width.

This conjecture proposes that bounded width is exactly the boundary between the simpler and harder Borel homomorphism problems. The surrounding discussion presents it as a possible scenario suggested by Thornton's work; its general status is not established here.