In the English Grammar Profile, B1 point 101 in MODALITY/adverbs is defined:
‘could’ + subject + ‘possibly’ to make requests more polite
*This is a very rare structure across corpora for a B1 structure. This point overlaps the more common and general: B1 questions with adverbs:
Could you possibly tell me how to fix this?
An iWeb search for:
. Could I|you possibly *
1 . COULD YOU POSSIBLY TELL 34
2 . COULD YOU POSSIBLY GIVE 26
Could you possibly give me a ride to Las Vegas?
3 . COULD YOU POSSIBLY HELP 24
4 . COULD I POSSIBLY BE 20
5 . COULD YOU POSSIBLY MAKE 20
6 . COULD YOU POSSIBLY DO 16
7 . COULD YOU POSSIBLY EXPLAIN 15
8 . COULD I POSSIBLY HAVE 14
9 . COULD YOU POSSIBLY POST 13
10 . COULD YOU POSSIBLY SEND 13
‘Could’ to express possibility is also B1.
Note that this form is not most common with requests:
* Could _P possibly *
1 . COULD IT POSSIBLY BE 192
2 HOW COULD IT POSSIBLY BE 153
How could it possibly be bad for football to have the world‘s best teams come here, and for so many people to be interested?
3 HOW COULD YOU POSSIBLY KNOW 133
(what) 4 MORE COULD YOU POSSIBLY WANT 121
5 ? COULD IT POSSIBLY BE 85
6 WHAT COULD IT POSSIBLY BE 80
7 COULD IT POSSIBLY BE 79
8 HOW COULD I POSSIBLY BE 68
9 , COULD IT POSSIBLY BE 63
10 MORE COULD YOU POSSIBLY ASK 61
There is not one example of this grammar in the PELIC or TLC.