The mod-3 power-sum conjecture for real multisets

Let $X$ be a multiset of $n$ real numbers, and define its power sums by

$$\varpi_k(X)=\sum_{q\in X}q^k.$$

Mod-3 power-sum conjecture. A real multiset $X$ of cardinality $n$ is uniquely determined by the $n$ values

$$\varpi_{3k-2}(X),\qquad k=1,2,\ldots,n.$$

The preceding lemma establishes the analogous statement using the first $n$ consecutive power sums, while this conjecture asks whether the power sums with exponents congruent to $1$ modulo $3$ suffice instead. Its status is not resolved in the supplied source context.