The site's wording drops the condition, assumed by the problem's sources, that the moduli are distinct; the literature uses the bare term both ways. If repeated moduli are allowed, the answer is trivially yes: for every $B$, the $M$ residue classes modulo any single $M>B$ cover the integers. This trivial cover is the corpus's own observation, and no result about the site's wording is recorded. The change inserts "with distinct moduli"; nothing else changes. The evidence is the literature's statement of the problem as Erdős's. Nielsen, A covering system whose smallest modulus is 40 (J. Number Theory 129 (2009), abstract, p. 1 of the author version), a construction that settles nothing: "Paul Erdős, in 1950, asked whether for each positive integer $N$ there exists a finite set of congruence classes, with distinct moduli, covering the integers, whose smallest modulus is $N$." Hough [Ho15], §1, printed p. 361: "From [4], the minimum modulus problem asks whether there exist distinct covering systems for which the least modulus is arbitrarily large", where [4] is Erdős's 1950 paper and a distinct covering system is a finite collection of congruences $a_i\bmod m_i$ with $1<m_1<m_2<\cdots<m_k$ covering every integer. [BBMST22], abstract, printed p. 378: "Erdős asked if the moduli can be distinct and all arbitrarily large"; their §1 (printed p. 378) defines a covering system as any finite collection of arithmetic progressions that covers the integers, so in their usage the term alone does not carry distinctness. Hough and BBMST state the problem apart from their theorems' hypotheses, and the site's commentary, which credits Hough's bound $10^{16}$, the bound $616000$ and Owens's cover with minimum modulus $42$, fits only the distinct reading. Erdős's 1950 paper also prints the distinctness: its conjecture on p. 120 concerns systems of congruences $a_i\pmod{n_i}$ with $n_1<n_2<\cdots<n_k$ that cover every integer (result page Erdős 1950, p. 120). Whether the site's sources [Er55c] to [Er97e] print it is not recorded here.