σ-collectionwise Hausdorffness at singular strong limit cardinals

Assuming the Singular Cardinals Hypothesis, we prove the following property: σ-CWH:For every singular strong limit cardinal κ and ↗-normal space X such that for some χ<κ, every x∈X has a neighborhood base of size ⩽χ, if every closed discrete subspace of size <κ is σ-separated, then so is every closed discrete subspace of size κ. So for getting a model of the negation of σ-CWH, we require a large cardinal.