Bluetooth keyboard randomly disconnects when bluetooth headphones are connected

This has been insanely difficult to debug. The symptom:

When my Thinkpad bluetooth keyboard is connected via bluetooth and my bluetooth Bose QC 35ii's are connected via bluetooth, I will sometimes see the keyboard stop working:

This doesn't always happen
When it does happen, the keyboard only disconnects after you have not pressed a key for > 1 second (?). After that, it feels like the keyboard has gone to sleep: it will take a few seconds to wake up and re-pair with the laptop.
- I have not seen anything obvious in journalctl or dmesg, blueman-applet, bluetoothctl when the keyboard stops working. After tapping some keys on the keyboard, it will re-pair, and I see the following appear in bluetoothctl:

[CHG] Device 90:7F:61:0C:C8:BF Connected: no
[CHG] Device 90:7F:61:0C:C8:BF Connected: yes

So it's almost as if the laptop doesn't even realize the keyboard has disconnected until the keyboard reconnects?

What's weird is that I've never seen this happen in Windows, but I also use Windows significantly less than Linux....

Things I've tried:

Setting usb autosuspend to 5 seconds (as per https://wiki.archlinux.org/index.php/Power_management#USB_autosuspend). This doesn't seem to have any effect. The keyboard still sporadically disconnects, and it disconnects in less than 5 seconds after you last use it.
Nothing appears in the output of allprocs when this breaks. So, if this is something the laptop is doing, it's either something about the way the connection is created that's putting the keyboard in a weird state, or it's something "silently" happening with an existing process or part of the kernel?

Misc links:

jfly / dotfiles

Bluetooth keyboard randomly disconnects when bluetooth headphones are connected #13