An asphericity conjecture and Kaplansky problem on zero divisors

Suppose a group representation H = 〈A∥ℛ〉 is aspherical, nx ∉ A, W(A ∪ x) is a word in alphabet (A ∪ x)^±1 with nonzero sum of exponents on x, and the group H naturally embeds in G = 〈A ∪ x ∥ℛ ∪ W(A ∪ x)〉. It is conjectured that the presentation G = 〈A ∪ x ∥ℛ ∪ W(A ∪ x)〉 is aspherical if and only if G is torsion free. It is proven that if this conecture is false and G = 〈A ∪ x ∥ℛ ∪ W(A ∪ x)〉 is a counter example, then the integral group ring Z(G) of torsion free group G will contain zero divisors. Some special cases when this conjecture holds are also indicated.