The site's wording admits two degenerate readings. First, "mutually coprime" in its usual sense, every pair coprime, is met vacuously by the one-term list $2^n-1$, so $h(n)=2$ for every $n$: the smallest instance is $n=2$, where the collective gcd gives $h(2)=3$ and the pairwise reading gives $h(2)=2$. On that reading the three questions are trivial, and the site's own remark that $h(n)=n+1$ if and only if $n+1$ is prime fails at every $n\ge2$ with $n+1$ prime. Second, the third question carries no quantifiers; read for one fixed $\epsilon$ and every $n$, it fails at $n=3$ for every $\epsilon<\log2/\log3$ ($P(3)=2$, $h(3)=3$) and at $n=86$ for every $\epsilon<\log3/\log86$ ($P(86)=3$, $h(86)=5$, the example Zeng's listing gives), where $P(n)$ is the greatest prime $p$ with $p-1\mid n$. Both values of $h$ were checked by direct computation of the gcds. The change replaces "mutually coprime" by "relatively prime, that is, $\gcd(2^n-1,3^n-1,\ldots,h(n)^n-1)=1$" and inserts "for every $\epsilon>0$ and all sufficiently large $n$" in the third question; nothing else changes. The evidence is the poser's own text, Erdős (1974), Part II. On printed p. 199 Erdős defines $h(n)$ as "the smallest integer for which the numbers $\{2^n-1,3^n-1,\ldots,h(n)^n-1\}$ are relatively prime" and says at once that $h(n)=n+1$ when $n+1$ is prime and conversely, which holds only for the collective gcd; the unnumbered lemma on the same page, "The set of integers $k^n-1$, $2\le k\le n+1$ is relatively prime", uses the phrase for the collective gcd, since for $n\ge3$ the list contains $2^n-1$ and $4^n-1$, and the first divides the second. The site's commentary states the same equivalence. On printed p. 200 the third question reads "It is possible that if $A(n)$ is large (say $>n^\varepsilon$) then $A(n)=h(n)$": the hypothesis is that $A(n)=P(n)$ be large, with $n^\epsilon$ for a fixed $\epsilon>0$ as the example of largeness, so the statement concerns large $n$. The first defect is the site's ("mutually coprime" for Erdős's "relatively prime"); the missing quantifiers are already in Erdős's text. The formal-conjectures statement file, which counts with the site, reads the problem the same way. No result about either degenerate reading is recorded.