[Feature Request] Auto-save conversation to project-local markdown files before compaction

Could compaction automatically write the about-to-be-compacted conversation to readable markdown files in the project directory?

Proposed structure:

{project}/.claude/prompt-history/
  20260218-1430 - Discussion on terminal UI.md
  20260218-1545 - Implementing validation hooks.md

Each file:

• {YYYYMMDD-HHMM} - {auto-generated description}.md
• Readable markdown with user/assistant turns
• Project-local (travels with the project, not stored globally)

Why project-local: Conversations are project-specific. Storing in .claude/prompt-history/ keeps history co-located with the work and avoids mixing conversations from different projects.

How this differs from existing proposals:

• #13239 — stores globally in ~/AI_claude/hist/, not project-local
• #17428 — UUID-named files in ~/.claude/, not human-browsable
• #13714 — manual /export, not automatic at compaction

If PreCompact hooks (#17237) ship first, this could be built as a hook — but descriptively-named, browsable conversation segments would be valuable as a built-in feature.

Related: #17237, #14258, #16300, #21188