In a pre-election period, candidates may, in the course of the public political campaign, adopt a strategic behavior by modifying their advertised political views, to obtain a better outcome in the election. This situation can be modeled by a type of strategic candidacy game, close to the Hotelling-Downs framework, which has been investigated in previous works via political views that are positions in a common one-dimensional axis. However, the left-right axis cannot always capture the actual political stances of candidates. Therefore, we propose to model the political views of candidates as opinions over binary issues (e.g., for or against higher taxes, abortion, etc.), implying that the space of possible political views can be represented by a hypercube whose dimension is the number of issues. In this binary strategic candidacy game, we introduce the notion of local equilibrium, broader than the Nash equilibrium, which is a stable state with respect to candidates that can change their view on at most a given number of issues. We study the existence of local equilibria in our game and identify natural conditions under which the existence of an equilibrium is guaranteed. To complement our theoretical results, we provide experiments to empirically evaluate the existence of local equilibria and their quality.