Unified (or cross-readable) session store across Claude Code CLI and the Claude Desktop app

Problem

The Claude Code CLI and the Claude Desktop app maintain separate session stores. /resume in the Desktop app lists only CLI sessions (the ~/.claude .jsonl transcripts); Desktop-app sessions (tracked separately, e.g. a HOST_SESSION_ID local_…) do not appear and cannot be resumed via /resume.

Impact

• A conversation started in the Desktop app cannot be resumed — or reset via the "Resume from summary" flow — because that flow reads the CLI store, which never contains the Desktop session.
• The cheap background resume-summary reset is effectively unavailable to Desktop-app users.
• No portability between surfaces: work started in one can't be picked up in the other.

Proposed behavior

Either a shared session store both surfaces read/write, or a cross-read bridge so /resume (and resume-from-summary) can list and load sessions regardless of which surface created them. Same underlying engine already; this is a bookkeeping-unification problem, not an engine one.

Verification note

Observed directly: running /resume in the Desktop app surfaced only CLI sessions and not the active Desktop session.