The initiative field is available in the Jira integration under the Aha! feature record. It works as expected when initially sending the Aha! feature to Jira. (Note: this is not the use case of mapping the initiative record.)
However, the mapped initiative field is not automatically updated in Jira when the text of the initiative name is changed in Aha!
The use case is just changing the text of the existing initiative name, not changing which initiative is associated with the feature.
Currently, need to manually select "Resend all fields" from each feature record in Aha! to update the initiative field in Jira.
This would apply to any record where the initiative field can be mapped in the integration.