Let $E$ be a nowhere dense subset of $\mathbb{R}\times \mathbb{R}$. For $x\in \mathbb{R}$, define $$E_x=\{ y\in\mathbb{R}\mid (x,y)\in E\}.$$ Let $D$ denote the set of $x$ for which $E_x$ is NOT nowhere dense in $\mathbb{R}$. By the Kuratowski-Ulam Theorem, we know that $D$ is of first cateogory in $\mathbb{R}$. My question is: can we strengthen this conclusion by saying that $D$ is nowhere dense in $\mathbb{R}$? If not, is there a counterexample?

Thanks in advance!