Degree-three invariant conjecture for generic orbit recovery in symmetric-group representations

Let $S_n$ act on $V=(\mathbb{C}^n)^d$ by permuting the coordinates in each copy of $\mathbb{C}^n$. The invariants of degree at most three are the multisymmetric power sum polynomial invariants in the coordinate variables, and they are said to list resolve the generic orbit when they contain a transcendence basis for $\operatorname{Frac}(\mathbb{C}[V]^{S_n})$.

Degree-three invariant conjecture. The invariants of degree at most three list resolve the generic orbit in $(\mathbb{C}^n)^d$ if and only if

$$\frac{1}{6}(d^3 + 6d^2 + 11d) \geq nd.$$

The inequality counts the available power sum multisymmetric invariants of degrees at most three and is therefore a necessary dimension condition for them to contain a transcendence basis. The conjecture is motivated by numerical Jacobian computations for small symmetric groups and values of $d$, but its general validity is not established in the supplied text.