How to flash a factory image onto a Nexus 5x with locked bootloader?

This happened today:

1. I was using my Nexus 5X when it suddenly started rebooting.


2. It got stuck in a bootloop. It would never boot past the Google logo and I couldn't turn it off.


3. I was able to enter fastboot mode and even recovery mode from there.


4. I wiped the phone from recovery mode which didn't help.


5. I left the phone lying with the fastboot menu open because I thought maybe it is overheated and will cool down (it was slightly warm).


6. When I returned, the phone was off. I started it and this time it went past the Google logo up to the Android logo (with the animation). However, after displaying "Android" it would just power off.


7. I tried to boot it a few times but it would never go past the Android logo. Eventually the bootloop started happening again.


8. I am now back to step 2 except I can't enter recovery mode any more. If I choose recovery mode from the fastboot menu, I never see the lying Android figure with the exclamation mark.

I found many tutorials about fixing bootloops on the Nexus but they all seem to require an unlocked bootloader.

The bootloader on the device is locked. I can see the device in fastboot devices but I can't unlock it.

Any idea how to repair this? Can I somehow flash a factory image onto the device even though OEM unlock is disabled? There is no data on the device that would need to be preserved. I'm ideally looking for a Linux-based solution.