Converse implication for topological bounded shadowing

Determine whether every bijective map of a metric space with the topological bounded shadowing property necessarily has the topological shadowing property.

Background

The paper introduces topological bounded shadowing by requiring only bounded pseudo-orbits to be shadowed, with the accuracy and pseudo-orbit tolerances given by positive continuous functions. It is immediate that topological shadowing implies topological bounded shadowing because every bounded pseudo-orbit is a pseudo-orbit.

The unresolved question is whether this implication can be reversed for bijective maps of metric spaces. Establishing the converse would show that restricting attention to bounded pseudo-orbits does not weaken the shadowing property in this topological, variable-tolerance setting.