Which tmux feature exhibits an unresolved behavior regarding environment variable inheritance across panes and sessions?

Goal

Understand the current state of environment variable propagation in tmux when creating new panes, synchronizing input, or detaching and reattaching sessions.

Constraints & Uncertainty

The documented set -g default-command option assigns a command to new panes, yet it does not automatically carry over environment variables set in the parent session or earlier panes. The synchronize-panes feature broadcasts input to all panes in a window but does not propagate changes to environment variables made in one pane to the others. Additionally, detaching a session and reattaching from a different client retains the original client’s environment, but any variables exported by commands executed within the session are not reflected in new clients.

Specific Questions

1. Does set -g default-command inherit the current session’s environment when spawning a new pane, or must variables be exported explicitly in each pane’s shell?

2. Is there a documented or recommended approach to propagate environment variable changes made in one pane to all other panes when synchronize-panes is enabled?