[Bug] Agent Teams split-panes mode disabled in VSCode terminal with tmux

Bug Description
Agent Teams split-panes mode stopped working in VSCode integrated terminal after VSCode update.

Environment:

• Running inside tmux session (TMUX env var is set)
• TERM_PROGRAM=vscode
• Agent Teams enabled (CLAUDE_CODE_EXPERIMENTAL_AGENT_TEAMS=1)

Expected: When spawning a teammate for another project (e.g., "project:aicluster"),
a new tmux split-pane should open automatically with size adjustment (as it did before).

Actual: No split-pane is created. The teammate mode "auto" may be incorrectly
detecting VSCode terminal and disabling split-panes, even though tmux is active.

Possible cause: TERM_PROGRAM=vscode detection may override tmux detection,
preventing split-panes mode from activating.

Environment Info

• Platform: linux
• Terminal: vscode
• Version: 2.1.81
• Feedback ID: 56186467-b791-4352-bd1f-658bf5ba1b20

Errors

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