I've been told quite recently that your log book is your
personal log and it is your responsibility to maintain it. The details are yours to do with what you please and if that includes transfering them to a bright pink fluffy book with tassles then that is your choice - but you must have one!
In this case, the signatures are irrelevant and only act as a reminder of who actually instructed you.
IMO,
not transferring the information to a new log book would be more of an issue for the aviation authorities than losing a few signatures.
Good luck in the interview
Cheers
Sparky