I asked Alexis Virelizier (coauthor of the book mentioned above) by e-mail. Here is his answer (reproduced with his authorisation):
The answer is yes. See Theorem 1.1 (page 2291) in the following paper by Funar: https://www-fourier.ujf-grenoble.fr/~funar/2013geomtopol.pdf
The answer is yes. See Theorem 1.1 (page 2291) in the following paper by Funar: https://www-fourier.ujf-grenoble.fr/~funar/2013geomtopol.pdf