Runs the organization's Support Sidekick agent synchronously against the
ticket and returns a suggested reply. The agent runs in read-only mode,
so it cannot mutate the ticket; the returned content is only a suggestion
for the requesting agent to review and send via the messages endpoint.

Four modes are supported, chosen by which fields you send:

• Neither draft nor adjustment — fresh reply based on the ticket context.
• adjustment only — fresh reply that follows the provided direction.
• draft only — produces an improved version of the draft.
• draft + adjustment — rewrites the draft per the adjustment instruction.
  Empty or whitespace-only strings are treated as absent.