VS Code extension: Persist conversation sessions across window reload

Problem

When the VS Code window is reloaded (Ctrl+Shift+P → Reload Window, crash, power outage, or accidental close), the entire Claude Code conversation is permanently lost. There is no way to resume or restore it.

The CLI version supports --continue and --resume for session persistence, but the VS Code extension has no equivalent functionality.

Impact

• Any accidental window reload destroys hours of conversation context — this is extremely frustrating and feels like a critical oversight
• No protection against crashes or power outages
• Users must re-explain complex multi-step tasks from scratch
• The memory system helps but doesn't replace full conversation history
• This creates one of the worst user experiences possible — losing all work context without warning

Expected behavior

• Sessions should automatically persist across window reloads (saved to disk)
• A "Resume last session" or "Show session history" command in the VS Code command palette
• Feature parity with CLI's --continue / --resume functionality
• At minimum: auto-save conversation state so a simple reload doesn't destroy everything

Steps to reproduce

1. Open Claude Code in VS Code extension
2. Have a long conversation with multiple tool calls, code changes, context
3. Press Ctrl+Shift+P → "Reload Window" (or experience a crash/power outage)
4. Claude Code panel resets — entire conversation gone, no way to recover

Environment

• Claude Code VS Code Extension
• Windows 10 Pro
• Claude Code CLI 2.1.80 (has --continue/--resume, extension doesn't)

Priority justification

This should be a top priority fix. Session persistence is a fundamental feature for any development tool. Developers routinely reload VS Code windows, and losing all AI conversation context each time makes the extension unreliable for serious work. The CLI already solves this — the extension just needs parity.