Removing CH from the non-plasticity of small dense subsets of the real line

Determine whether the statement of Corollary 4 holds in ZFC without assuming the Continuum Hypothesis, namely: ascertain whether every dense subset X of the real line with cardinality |X| < 𝔠 is not plastic.

Background

Corollary 4 states that, under the Continuum Hypothesis (CH), every dense subset X ⊆ ℝ with |X| < 𝔠 is not plastic. This conclusion uses Theorem 1, which shows that every countable dense subspace of any normed space is not plastic; under CH, |X| < 𝔠 implies X is countable.

It remains unclear whether the CH assumption can be eliminated while retaining the same non-plasticity conclusion for dense subsets of ℝ of cardinality less than the continuum.