Research on influence maximization has often to cope with marketing needs relating to the propagation of information towards specific users. However, little attention has been paid to the fact that the success of an information diffusion campaign might depend not only on the number of the initial influencers to be detected but also on their diversity w.r.t. the target of the campaign. Our main hypothesis is that if we learn seeds that are not only capable of influencing but also are linked to more diverse (groups of) users, then the influence triggers will be diversified as well, and hence the target users will get higher chance of being engaged. Upon this intuition, we define a novel problem, named Diversity-sensitive Targeted Influence Maximization (DTIM), which assumes to model user diversity by exploiting only topological information within a social graph. To the best of our knowledge, we are the first to bring the concept of topology-driven diversity into targeted IM problems, for which we define two alternative definitions. Accordingly, we propose approximate solutions of DTIM, which detect a size-k set of users that maximizes the diversity-sensitive capital objective function, for a given selection of target users. We evaluate our DTIM methods on a special case of user engagement in online social networks, which concerns users who are not actively involved in the community life. Experimental evaluation on real networks has demonstrated the meaningfulness of our approach, also highlighting the opportunity of further development of solutions for DTIM applications.