" Do you think you ____ open the door for me?"
If somebody had two broken arms in plaster of paris casts, then I might say
" Do you think you can manage to open the door for me?"
Other than that, he of course has the ability to open a door, so why would I ask him if in fact he does have the ability? - Can you open...?
'could' is used in these situations, when we are making a polite request of someone, could I ask them to take the trouble to help me by opening the door because, say, I am carrying a heavy tray of dishes.
Student or Learner