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Source. Conjecture 1, PDF p. 1 of arXiv:math/0604347v2.

Statement

Let $k\geq2$. If the $k$ congruence classes

$$a_i\pmod{m_i}\qquad(1\leq i\leq k)$$

are pairwise disjoint, then there are indices $i<j$ such that

$$\gcd(m_i,m_j)\geq k.$$

The paper (p. 1) attributes the conjecture to Z.-W. Sun, who posed it on the number theory listserver in May 2003, and abbreviates it DCCC. It notes that the $k$ classes $1\pmod k,2\pmod k,\ldots,k\pmod k$ show the bound $k$ cannot be raised, and it recalls that $k=2$ is the Chinese remainder theorem and that $k=3$ follows from the pigeonhole principle (Graham, pp. 1--2).

Proof scope. This is Sun's conjecture as stated by O'Bryant. Theorem 3 proves only the finite range recorded there.

Bears on. Problem 202: the conjecture concerns the moduli of any family of pairwise disjoint congruence classes, including the families with distinct moduli that problem counts, but it says nothing about that problem's maximum number of classes with distinct moduli at most $N$.