Let $\mathcal{A}$ be a central hyperplane arrangement in a (finite dimensional) real vector space $V$. Assume for each hyperplane $H\in\mathcal{A}$ that we're given a labelling $H^+$, $H^-$ of the connected components of $V\setminus H$.

Given a subset $\mathcal{B}\subseteq \mathcal{A}$, is it possible for the set $$ \bigcap_{H\in \mathcal{B}} H^+ \cap \bigcap_{H\in \mathcal{A}\setminus\mathcal{B}} H^- $$ to be empty while the set $$ \bigcap_{H\in \mathcal{B}} H^+ \cap \bigcap_{H\in \mathcal{A}\setminus\mathcal{B}} (H\cup H^-) $$ is not?

If 1. is false in general, what if we restrict to the following case?: Let $C$ be a strongly convex full-dimensional polyhedral cone in $V$. Set $\mathcal{A}=\{\mathbb{R}F\mid F\text{ a facet of }C\}$, and for each $H\in \mathcal{A}$, let $H^+$ be the connected component of $V\setminus H$ containing the interior of $C$.