Status line model display shared across instances, shows wrong model

Description

When running multiple Claude Code instances on the same system with different models selected, the status line model indicator shows the wrong model. For example, one instance may show "Sonnet 4.5" in the status line even though /model confirms "Opus 4.6 (Default)" is the active model for that session.

Steps to Reproduce

1. Open two separate Claude Code instances (e.g., in different terminal tabs or projects)
2. Select different models in each instance (e.g., Opus 4.6 in one, Sonnet 4.5 in another)
3. Observe the status line in both instances

Expected Behavior

Each instance's status line should reflect the model selected in that instance.

Actual Behavior

The model shown in the status line appears to be shared/global state across all instances on the system, so whichever instance updated last "wins" and all instances show that model regardless of their actual selection.

Environment

• Platform: Linux (Pop!_OS / Ubuntu-based)
• Claude Code CLI