  This paper is a part of the series proving the Gaiotto conjecture for basic
classical quantum supergroups. The previous part arXiv:2107.02653 [math.RT] ,
arXiv:2306.09556 [math.RT], proved the Gaiotto conjecture for the general
linear quantum supergroups $U_q(\mathfrak{gl}(N|M))$. Here we deal with the
exceptional quantum supergroup $U_q(\mathfrak{f}(4))$.
