Implement DB-based conversation message storage (engine-agnostic) that
works across both Claude API and Agent SDK engines. Add ChatGPT/Claude-style
conversation history drawer in Flutter with date-grouped session list,
session switching, and new chat functionality.
Backend: entity, repository, context service, migration 004, session/message
API endpoints. Flutter: ConversationDrawer, sessionId flow from backend
response via SessionInfoEvent, session list/switch/delete support.
Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>