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Source. Conjecture 4, PDF p. 2 of arXiv:math/0604347v2.

Statement

Interpret the statement for an integer $k\geq2$. Suppose $A_1,\ldots,A_k$ are pairwise disjoint left cosets of subgroups $H_1,\ldots,H_k$ in a group $G$. Then there are indices $i<j$ such that

$$\gcd([G:H_i],[G:H_j])\geq k.$$

The printed statement does not separately say $k\geq2$, but its conclusion requires a pair $i<j$, so that range is implicit and is made explicit here. It also does not separately say that the subgroup indices are finite, although its displayed greatest common divisor uses them as ordinary integers. This page preserves that second implicit convention rather than silently adding a finite-index hypothesis.

Proof scope. This is a conjecture attributed to Sun, not a proved theorem in this source.