A basis for the equational identities of unordered $n$-tuples

Question

This is somewhat related to my previous question, here: Unordered $n$-tuples can't define unordered $(n+1)$-tuples. As before, suppose we are working in the model theory of ZFC set theory. Also as before, for every positive integer $n$, I define an $n$-ary operation $f_n$, such that $f_n(x_1,...,x_n)=\{x_1,...x_n\}$. Now, fix a positive integer $n$, and consider the identities of the structure $(V;f_n)$, where $V$ is a model of ZFC set theory. What is the basis of the equational identities (in the sense of universal algebra) of that structure? I conjecture that the "permutative" identities are enough. An $n$-ary operation is said to be permutative if any permutation of its variables is equal to the original operation. So, to give examples, a binary operation $f$ is permutative if and only if it is commutative, and a ternary operation $g$ is permutative if and only if it satisfies the identities $g(x,y,z)=g(y,x,z)=g(x,z,y)=g(y,z,x)=g(z,x,y)=g(z,y,x)$. Is this conjecture true? If so, I want a proof. If not, is there still a finite basis for the equational identities of that structure?