A delegated subagent child (source='subagent' in state.db) usually has no WebUI
sidecar but is registered in the WebUI index sharing the parent's lineage. That
made GET /api/session -> _claim_or_synthesize_cli_session return 'was_webui' ->
404, so the child pane opened empty despite state.db holding messages.
- api/routes.py: add _state_db_session_source() + _is_subagent_child_session_id();
exclude subagent children from the was_webui 404 gate so they recover their
state.db transcript (the #2782 self-heal 404 for deleted WebUI sessions is kept).
- static/sessions.js: add _isSubagentChildSession() + _sessionNeedsServerImportForLoad()
(kept separate from _isExternalSession to avoid widening refresh-gating); the
main session-tap, lineage-segment, and child-open handlers now trigger the
import/merge path for subagent children.
- tests: test_5307_subagent_child_transcript.py (7 tests).
Fixes#5307