Background process reminders show stale status after conversation compression

Description:

Background Bash processes that complete before conversation compression continue generating reminders with stale status information after the conversation is resumed.

Steps to Reproduce:

1. Start a background Bash command in a conversation
2. Background process completes successfully
3. Conversation reaches context limit and gets compressed/summarized
4. Continue conversation from summary
5. System continues sending reminders: "Background Bash <id> (status: running) Has new output available"
6. Checking with BashOutput shows: "status: completed" with exit code 0
7. Attempting KillShell returns: "Shell <id> is not running, so cannot be killed (status: completed)"

Expected Behavior:

• Background process completion status should survive conversation compression
• OR completed processes should be excluded from the resumed conversation state
• Reminders should stop once process completes

Actual Behavior:

• Reminders persist with stale status ("running" instead of "completed")
• Reminder system appears to have cached pre-compression state
• Actual process status is correct when checked, but reminders don't update
• No way to clear or dismiss the stale reminders

Root Cause Hypothesis:
The conversation compression/summarization process may not properly serialize background process completion state, causing the reminder system to revert to or cache the pre-completion state.

Environment:

• Process started in original conversation, completed before compression
• Conversation was summarized/compressed due to context limits
• Shell ID: 5f8608
• Confirmed completed via BashOutput tool (exit code 0, status: completed)
• Confirmed completed via KillShell error message

Impact:

• Creates persistent noise in resumed conversations
• Shows incorrect status information
• Can lead to wasted tool calls checking already-completed processes
• May affect long-running sessions that frequently hit context limits

Suggested Fixes:

1. Ensure background process completion state is properly serialized in conversation summaries
2. Clear completed process reminders during compression
3. Provide a tool to manually dismiss stale reminders
4. Add state reconciliation on conversation resume to sync reminder status with actual process status