Enhancement: list children of a specific toolbox instance

I would like to be able to list the child processes that are running in a specific toolbox instance.

My use case is as follows: I run tmux on my host (silverblue). The tmux spawns a bunch of toolbox shells. I would like to have a tmux binding that creates a new toolbox shell in the working directory of my active pane. Usually I could use tmux's internal variables for this, but because the child of tmux is toolbox rather than the shell within toolbox, the cwd of that never changes. I also can't switch to just running tmux inside of toolbox, since I want to be able to open a non-toolbox shell within the tmux instance to run commands on the host.

If I could, given the PID of a 'toolbox enter' process, get a list of processes within that (or just the one that has control of the terminal), I could read that process' working directory from a script and change to it before running a new 'toolbox enter'. A bit convoluted, perhaps.

At the moment simply walking through /proc finding child processes doesn't work, since that tree ends at the 'conmon' process (which may or may not be defunct, it seems).