Does unbounded width imply linear width for PCSPs?

Ascertain whether every promise constraint satisfaction problem PCSP(A, B) with unbounded width necessarily has linear width; equivalently, determine if the converse of the known implication for CSPs (that unbounded width implies linear width) extends to the promise setting.

Background

The paper discusses width as a measure of the power of the local-consistency algorithm. For CSPs, it is known that unbounded width is equivalent to linear width, but for PCSPs only the direction from linear width to unbounded width is currently established.

The authors explicitly note that extending the converse direction to PCSPs is not known, highlighting a gap in the understanding of width behavior for promise problems.