A set M of nonzero integers is said to split a finite abelian group G if there is a subset S of G for which M • S = G\{0] . If, moreover, each prime divisor of |G| divides an element of M, we call the splitting purely singular. It is conjectured that the only finite abelian groups which can be split by {1, ... , k} in a purely singular manner are the cyclic groups of order 1, k + 1 and 2k + 1 . We show that a proof of this conjecture can be reduced to a verification of the case gcd(|G|, 6) = 1 .