This is a special case of a question that has not been answered so far.

If $(X,\tau)$ is a connected $T_2$ space with more than 1 point, we define its *nowhere dense covering number* $\nu(X)$ by the smallest cardinality that a partition of $X$ into nowhere dense subsets of $X$ can have.

**Question.** Is there a connected Hausdorff space $(X,\tau)$ such that $\nu(X) = 3$?