Gaiotto Conjecture for $\mathrm{Rep}_q(\mathrm{F}(4))$

Abstract: 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 $U_q(\mathfrak{f}(4))$.