Strict separation between bounded chaining levels

Construct a measure-class preserving action of a countable group that is $(k+1)$-chaining but not $k$-chaining for some $k$, or at least construct one that is boundedly chaining but not 1-chaining.

Background

The authors construct singular actions separating successive chaining levels and later obtain examples separating successive levels of essential chaining for nonsingular actions. They explicitly state that analogous examples for ordinary measure-class preserving actions are not known, leaving open whether bounded chaining collapses to 1-chaining or whether successive bounded levels can be separated.