Set theory and cyclic vectors

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Let H be a separable, infinite dimensional Hilbert space and let S be a countable subset of H. Then most positive operators on H have the property that every nonzero vector in the span of S is cyclic, in the sense that the set of operators in the positive part of the unit ball of B(H) with this property is comeager for the strong operator topology. Suppose κis a regular cardinal such that κ≥ ω₁ and 2<κ = κ. Then it is relatively consistent with ZFC that 2ω= κand for any subset S ⊂ H of cardinality less than κthe set of positive operators in the unit ball of B(H) for which every nonzero vector in the span of S is cyclic is comeager for the strong operator topology.