Topological Erdős similarity conjecture for uncountable sets

Let $P$ be a set in the relevant Euclidean space, and let $A$ be a Baire set. An affine image of $P$ is a set obtained from $P$ by an affine transformation. Topological Erdős similarity conjecture. For each uncountable set $P$, there exists a non-meager Baire set $A$ that contains no affine image of $P$. This is posed as the direct topological analogue of Erdős's measure-theoretic similarity conjecture; the paper establishes universality for all bounded countable sets, leaving the uncountable case as the stated conjecture.