Option to hide background task count from status bar

Description

When using Claude Code with background tasks (e.g., file watchers started via run_in_background), a row appears below the custom status line showing the count of running background Bash tasks. There is currently no way to hide or suppress this row.

Use Case

I run multiple persistent background watchers (file system watchers, watchdogs) that are always active during a session. The background task count indicator provides no actionable information in this workflow — I already know the watchers are running. It adds visual clutter to the status area, especially when running 8+ concurrent sessions where every pixel of screen real estate matters.

Requested Behavior

Add a setting (e.g., statusLine.showBackgroundTaskCount in ~/.claude/settings.json) to hide the background task count row from the status area. The background tasks themselves should continue to function normally — only the UI indicator would be suppressed.

Alternatives Considered

• Custom StatusLine hook: The StatusLine hook only controls the text of the custom status line — it cannot suppress other UI elements rendered by Claude Code.
• Reducing background tasks: Not viable — the watchers are essential to the workflow.

Environment

• Claude Code v2.1.x on Windows 11
• Multiple concurrent sessions with 2-3 background tasks each

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Session: Mouse (c877)