Logger Heads
Nope!
See Lasors P56. Pilot-in-Command Para. 1(c).
You may, in a single pilot aeroplane, log P1(s) only when the flight is a licensing test and the examiner is acting as such.
Club check rides are not a licensing requirement. It is an occasion when the hirer is protecting their own interest. Most clubs will nominate an instructor to supervise a check ride. You are, are you not, checking that the pilot has current satisfactory skill which includes their PIC skills.
The PPL holder may if no intervention is required log the flight P1 but so may the instructor in accordance with the above section from LASORS 2007