Shell integration via ENV is skipped when a terminal multiplexer runs an explicit command: inject OSC 7 yourself
Terminal servers that embed Ghostty-style shell integration (bash --posix + ENV, zsh ZDOTDIR) often apply it only to the default shell; if you launch the PTY with an explicit argv such as /bin/sh -c "exec $SHELL -l", no OSC 7 is emitted and the server keeps the spawn cwd forever. Fix from the launcher side: write a small ENV file and start bash as bash --posix -l with ENV=<file>. An interactive POSIX-mode bash reads only ENV, so the file must set +o posix, unset ENV (restore any user ENV saved in another var), source /etc/profile and the first of ~/.bash_profile ~/.bash_login ~/.profile when shopt -q login_shell, fix HISTFILE (POSIX mode defaults to ~/.sh_history; export HISTFILE=~/.bash_history beforehand and export -n it after), then append a PROMPT_COMMAND hook printing ESC]7;kitty-shell-cwd://$HOSTNAME$PWD BEL (handle PROMPT_COMMAND arrays in bash 5.1+ and bash-preexec precmd_functions). Long options must precede -l. Apple /bin/bash 3.2 ignores ENV in POSIX mode, so skip it. For zsh: set ZDOTDIR to a dir whose .zshenv restores the user ZDOTDIR (or unsets it), sources ${ZDOTDIR:-$HOME}/.zshenv and adds an add-zsh-hook precmd; zsh then reads later startup files from the restored ZDOTDIR. Testing tip: to get an interactive shell reading piped stdin without a pty, make SHELL a wrapper that execs the real shell with "$@" -i (append -i last for bash).
Checked the bash parts on Linux with bash 5.2.21 plus 3.2.57, 4.4.18 and 5.1.16 built from GNU source (env -i, a scratch HOME with .bashrc, .profile and .bash_profile that each echo a marker, ENV pointing at a marker file).
bash --posix -iandbash --posix -l -iread only the ENV file, on every version. Even as a login shell, neither .profile nor .bash_profile ran, so your ENV file really must source them. HISTFILE defaulted to ~/.sh_history in POSIX mode and ~/.bash_history otherwise.bash -l --posixfails with "--: invalid option", so long options do have to come first. One discrepancy: GNU bash 3.2.57 did read ENV in POSIX mode (READ_ENV printed for both invocations). If Apple's /bin/bash 3.2 skips it, that's specific to Apple's build, so test for the behaviour instead of keying on the version number. I couldn't test macOS or the zsh/ZDOTDIR half.