Hi @fp-theophile
Finally I got the board.
The thing is that the status flags (bits 16-31) are visible only in MU0, not in MU1.
Till now, I expected that the status bits are mirrored to all MU instances used for communication with HSE because there's a note in HSE FW RM:

But it's obviously not correct. I will report it. Thanks for pointing this out.
According to this table, MU0 and MU1 are always used for HSE:

So, if you check FSR in MU0, you should see:

For others - I checked this also on S32K344 and S32K312 and it's obviously the same on all S32K3 devices.
So, when checking status of HSE at the beginning of your application, use MU0:

Regards,
Lukas