Rigidity conjecture for generic symmetric hypergraphs

Let $n\geq 2$. Let $\mathscr{L}_n=\{R\}$ be a relational language with $R$ an $n$-ary relation, and let $\operatorname{AgeK}_{\mathrm{sym},n}$ be the class of all finite $\mathscr{L}_n$-structures in which $R$ is symmetric, meaning that

$$\bigwedge_{\tau\in\operatorname{Sym}([n])}(\forall x_0,\dots,x_{n-1})\,R(x_0,\dots,x_{n-1})\leftrightarrow R(x_{\tau(0)},\dots,x_{\tau(n-1)}).$$

For an uncountable cardinal $\kappa$, let $\mathbb{M}_\kappa$ be the unique structure in the empty language of size $\kappa$, and let $G$ be generic for the forcing $\operatorname{Fn}(\mathbb{M}_\kappa)[\operatorname{AgeK}_{\mathrm{sym},n}](\omega)$ over $V$. Rigidity conjecture for generic symmetric hypergraphs. The structure $\bigcup G$ is rigid in $V[G]$. This conjecture proposes that the forcing construction yields a symmetric generic hypergraph with no nontrivial automorphisms, despite the absence of function symbols that could name ordered tuples. The corresponding rigidity argument is immediate for sufficiently asymmetric relational structures but does not presently extend to symmetric $n$-hypergraphs, so the claim remains open.