The word "(essentially)" is Erdős's: [Er65b], printed p. 213, display (37), reports the conjecture as Turán's, told to Erdős in conversation, and leaves the word undefined; the site's commentary records that Erdős does not elaborate on what it may mean. Read as the site words it, with the conclusion that the $z_j$ are the $n$th roots of unity, the conjecture fails for every even $n$: $z_1=1$, $z_2=i$ gives $s_k=1+i^k$, which vanishes for every $k\equiv2\pmod4$ (Quanyu Tang, site thread, 20 September 2025), and for $n=2m$ the $m$th roots of unity together with their rotation by $e^{\pi i/(2m)}$ give $s_k=0$ for every $k\not\equiv0,m,3m\pmod{4m}$, a run of $n-1$ zeros in every period of length $4m$ (Tao Hu's construction, posted by Tang the same day). The site's curator, Thomas Bloom, resolves the word through Tijdeman's theorem. The commentary (page last edited 1 October 2025) states the conclusion as "if $n$ is odd then the $z_i$ must be exactly the $n$th roots of unity, and if $n$ is even they must be the vertices of two regular $(n/2)$-gons with the same circumscribed circle centred at the origin", credits Tijdeman [Ti66] and labels the problem PROVED (LEAN); replying to the $n=2$ example in the thread on 20 September 2025, Bloom wrote that it "is not a counterexample though", given how vaguely the problem is described; and the formal-conjectures statement erdos_974, which the site's Lean mark follows, concludes that configuration. The precise Statement replaces "(essentially) $z_j=e(j/n)$" by that conclusion and changes nothing else. Under it the problem is proved: Tijdeman [Ti66] proves the classification from two runs of $n-1$ vanishing power sums for pairwise distinct $z_i$, and a single run already forces the $z_i$ to be distinct and nonzero (Proposition 1 of Hu, Tang and Zhang in the thread, proved in both Lean files). Under the site's wording the answer is yes for odd $n$, where the two readings agree, and no for every even $n$, by the construction above, which has no claim page; its authors went on to treat the classification as the problem's resolution. Erdős's setting in [Er65b] also requires $|z_i|\le1$ for $2\le i\le n$, which the site's wording omits; neither answer changes, since the conclusion forces $|z_i|=1$ and the construction lies on the unit circle. Unread: Turán's own statement of the conjecture, and its statement in [Ti66].