In my first test I'm using the MoodleBox 3.13.0-dev1 (64bits) because this is the current version on the microsd card in my Raspberry Pi. Nicolas made this image version for testing the MoodleBox with Debian Bullseye. https://github.com/moodlebox/moodlebox/issues/213#issuecomment-903275347
My MoodleBox is running the current Moodle 3.11.3+ (20210924). The Moodle App 3.9.5 is installed on an iPhone, an iPad and an old Samsung Tab S2. Yes, I can connect to the MoodleBox without any problem on all three devices. And I'm able to get two courses saved locally in the app. I can get only two courses because my MoodleBox only uses the Moodle App plan "Free". https://moodle.com/app/#plans
When I try to get downloaded the third course the app shows an error message with a hint to the Moodle App plans.
I hope that this post answers your questions.
Best regards, Ralf