Characterization of the Group A₅ × A₅ × A₅ by the Set of Conjugacy Class Sizes

For a finite group G, we denote by N (G) the set of its conjugacy class sizes. Recently, the following question was posed: given any n ∈ ℕ and an arbitrary non-Abelian finite simple group S, is it true that G ≃ Sⁿ if G is a group with trivial center and N (G) = N (Sⁿ)? The answer to this question is known for all simple groups S with n = 1, and also for S ∈ {A₅, A₆}, where A_k denotes the alternating group of degree k, with n = 2. It is proved that the group A₅ × A₅ × A₅ is uniquely defined by the set N (A₅ × A₅ × A₅) in the class of finite groups with trivial center.