On the singular abelian rank of ultraproduct II₁ factors

We prove that, under the continuum hypothesis c=₁, any ultraproduct II₁ factor M= ∏ω Mₙ of separable finite factors Mₙ contains more than c many mutually disjoint singular MASAs, in other words the { singular abelian rank of} M, r(M), is larger than c. Moreover, if the strong continuum hypothesis 2ᶜ=₂ is assumed, then r(M) = 2ᶜ. More generally, these results hold true for any II₁ factor M with unitary group of cardinality c that satisfies the bicommutant condition (A₀'∩ M)'∩ M=M, for all A₀⊂ M separable abelian.