Does unbounded width imply linear width for PCSPs?

Ascertain whether every promise CSP (PCSP) with unbounded width necessarily has linear width; that is, determine if the analogue of the CSP result—"every CSP of unbounded width has linear width"—extends to the promise setting.

Background

The width of a (P)CSP measures the strength required by the local-consistency algorithm to solve instances. For CSPs, it is known that unbounded width implies linear width. However, the authors note that the corresponding statement for PCSPs is not established.

This question matters for understanding the landscape of promise CSPs: a positive answer would yield a dichotomy between bounded and linear width, while a negative answer would allow intermediate (super-constant but sublinear) widths in the promise setting.