Summary: A target set selection model is a graph \(G\) with a threshold function \(\tau : V (G) \to \mathbb{N}\) upper-bounded by the vertex degree. For a given model, a set \(S_0 \subseteq V (G)\) is a target set if \(V (G)\) can be partitioned into non-empty subsets \(S_0, S_1, \dots, S_t\) such that, for all \(i \in \{1, \dots, t\}\), \(S_i\) contains exactly every vertex \(v\) outside \(S_0 \cup \cdots \cup S_{i - 1}\) having at least \(\tau (v)\) neighbors in \(S_0 \cup \cdots \cup S_{i - 1}\). We say that \(t\) is the activation time \(t_\tau (S_0)\) of the target set \(S_0\). The problem of, given such a model, finding a target set of minimum size has been extensively studied in the literature. In this article, we investigate its variant, which we call TSS-time, in which the goal is to find a target set \(S_0\) that maximizes \(t_\tau (S_0)\). That is, given a graph \(G\), a threshold function \(\tau\) in \(G\), and an integer \(k\), the objective of the TSS-time problem is to decide whether \(G\) contains a target set \(S_0\) such that \(t_\tau (S_0) \geq k\). Let \(\tau^\star = \max_{v \in V (G)} \tau (v)\). Our main result is the following dichotomy about the complexity of TSS-time when \(G\) belongs to a minor-closed graph class \(\mathcal{C}\): if \(\mathcal{C}\) has bounded local treewidth, the problem is FPT parameterized by \(k\) and \(\tau^\star \); otherwise, it is NP-complete even for fixed \(k = 4\) and \(\tau^\star = 2\). We also prove that, with \(\tau^\ast = 2\), the problem is NP-hard in bipartite graphs for fixed \(k = 5\), and from previous results we observe that TSS-time is NP-hard in planar graphs and W[1]-hard parameterized by treewidth. Finally, we present a linear-time algorithm to find a target set \(S_0\) in a given tree maximizing \(t_\tau (S_0)\).