The hierarchicalMenu and connectHierarchicalMenu widgets are not searchable in any flavor of InstantSearch. This is because only the root-level facet values are available in the initial search response.
Deeper levels in the hierarchy (for example, lvl1, lvl2) aren’t retrieved until a refinement is applied.
Because those values aren’t available upfront, the widget can’t support searchable facet behavior. Making deeper levels searchable would require retrieving the additional hierarchical facet levels.
If this searchable behavior is essential to your use case, one solution would to use separate refinementList widgets instead of a hierarchical menu widget.
You could then customize your front-end to show the lower level refinement lists only after a refinement is chosen from the top level. Note that this level of customization is out-of-scope for our Support team).