The Leggett-Garg (LG) inequalities were formulated for testing the incompatibility between the classical worldview of macrorealism and quantum theory. By analogy with the Tsirelson bound for Bell inequalities, the maximal quantum violation of LG inequalities is commonly referred to as the temporal Tsirelson bound (TTB). Nevertheless, several studies have claimed that, unlike the Tsirelson bound, the TTB can be exceeded within quantum theory. In this work, we critically examine the reasoning underlying these purported TTB violations and show that the corresponding arguments suffer from conceptual shortcomings; once these issues are addressed appropriately, no violation of the TTB is obtained. In particular, we demonstrate that the proposed TTB violations rely on frameworks that go beyond the standard LG framework and therefore are of limited relevance for testing the incompatibility between macrorealism and quantum theory.