Unicity for representations of reduced stated skein algebras

We prove that both stated skein algebras and their reduced versions at odd roots of unity are almost-Azumaya and compute the rank of a reduced stated skein algebra over its center, extending a theorem of Frohman, Kania-Bartoszynska and L\^e to the case of open punctured surfaces. We deduce that generic irreducible representations of the reduced stated skein algebras are of quantum Teichm\"uller type and, conversely, that generic quantum Teichm\"uller type representations are irreducible.