On Rana's suggestion I checked the MC transmission QPD (C1:IOO-MC_TRANS_SUM). I found that the readout is almost zero when the MC is unlocked.
I unlocked the Mode Cleaner turning off the LSC control and disabling the autolocker. The QPD reads 0.014. It seems that there is no offset.
I also checked with the IR card around the photodetector and I didn't see any stray beam.