Bailey's ring-theoretic characterization of the Generalized Continuum Hypothesis

Let GCH denote the assertion that for every infinite cardinal $\kappa$, there is no cardinal properly between $\kappa$ and $2^\kappa$. For an infinite cardinal number $\kappa$, let $R$ be a commutative ring of cardinality $\kappa$, and let its cardinal Krull dimension be the supremum of the cardinalities of chains of prime ideals of $R$. Bailey's conjecture. GCH is true if and only if for every infinite cardinal number $\kappa$, there exists a commutative ring $R$ of cardinality $\kappa$ and cardinal Krull dimension strictly larger than $\kappa$. The claim concerns an undecidability phenomenon in ZFC: the source states that ZFC together with $\neg$GCH disproves it, while its validity under GCH is established in the surrounding argument.