The AppRun script is executed with the shell on the host system. To reduce compatibility issues, I think it would be best to not depend on bash being installed and make all shell script wrappers POSIX-compliant so that they can use #!/bin/sh as shebang, which even on Ubuntu is often dash instead of bash.
Would this be something worth considering? I have had this idea in the back of my head since this issue, but I have finally customized my entrypoint.sh script so much that it works with pure dash.