The site's wording does not say which density it asks for: "the density of $n$ ... is at least $1-\eta$" can ask that the set of such $n$ have an asymptotic density and that it be at least $1-\eta$, or only that its lower density be at least $1-\eta$, and the two readings differ for a set whose density need not exist. The change inserts "lower" before "density"; nothing else changes. Erdős printed the question without a qualifier. [Er80], Section 6, item 2, printed p. 107 (library card: Erdos 1980), asks whether "the density of integers $n$" for which the largest of $P(n+1),\dots,P(n+k)$ exceeds $n^{1-\epsilon}$ "is greater than $1-\eta$"; the same survey writes "lower density" where it means it, in a statement of the same shape (printed p. 96, Gallagher's theorem on integers of the form $p+2^{u_1}+\cdots+2^{u_r}$), so Erdős's text is consistent with either reading and fixes neither. The reading is the site's curator's. In the site's thread Will Sawin wrote (1 May 2026) that it is reasonable to read "density at least" as "lower density at least" and "density at most" as "upper density at most" rather than requiring a proof that the density exists, and Thomas Bloom replied the same day: "I agree; I think this is generally how Erdős used these terms. (When he was specifically curious about the existence of the density he was generally clear about this.) I've tried to update all the problem descriptions to reflect this, but missed this one." Bloom's evidence is Erdős's general usage, and Bloom states the reading as the one the site's problem descriptions are meant to carry. The formal-conjectures statement erdos_1201 reads the bound the same way, as a liminf of the counting ratio. Erdős's form differs from the site's in two further ways, the omission of $P(n)$ and "greater than" in place of "at least"; neither changes the answer under the precise Statement, as the Formulation shows. Under the precise Statement the answer is yes, pending acceptance: Chojecki's note of 30 April 2026, recorded on its claim page (Chojecki, 2026), deduces from the Matomäki--Radziwiłł theorem that the exceptional set has upper density tending to $0$ as $k$ grows. Under the natural-density reading the question is open: Terence Tao wrote in the thread (30 April 2026) that the problem "remains technically open because it was not established that the natural density of the set actually exists"; Chojecki's second note (1 May 2026), recorded on its page, gives the natural density $1-\rho(1/(1-\epsilon))^{k+1}$ only under an unproven correlation hypothesis it names LPD and only for $\epsilon<1/2$; through that reading it reaches the precise Statement only in that range and only under the hypothesis, which the 30 April note already covers, so it does not count toward the problem's standing; and the logarithmic-density results of Teräväinen that Tao cited settle neither reading.