Full measure universality conjecture

A subset $X\subseteq\mathbb{R}$ is full measure universal if every Lebesgue measurable subset of $\mathbb{R}$ whose complement has Lebesgue measure zero contains an affine copy of $X$. Full measure universality conjecture. There is no uncountable full measure universal subset in $\mathbb{R}$. This is proposed as the measure-theoretic dual of the topological Erdős similarity conjecture; the source discusses its possible independence from $\mathsf{ZFC}$ but does not report a resolution.